Practice · Triangle inequality bounds isosceles legs · Sensim Math

Variant 1 easy

Among triangles whose three side lengths add up to $12$ , how many isosceles triangles (at least two sides equal) have all three side lengths whole numbers?

Your answer

Show solution

1 · Understandwhat's really being asked

Count the isosceles triangles with whole-number side lengths and perimeter $12$ .

Givens

The three side lengths add up to $12$ .
The triangle is isosceles: at least two sides are equal.
Every side length is a whole number.

Unknowns

How many such triangles there are.

Constraints

Three lengths make a triangle only if any two of them together are longer than the third.

2 · Planchoose the strategy

#13 Convert to Algebra · also uses: #2 Make a Systematic List

Name the equal sides $x$ and the third side $y$ . The perimeter turns $y$ into an expression in $x$ , the triangle condition gives a lower bound on $x$ , and the third side being positive gives an upper bound. Then list the whole numbers in between.

3 · Execute4 carry out the plan

1Write the third side in terms of the equal sides

#13 Convert to Algebra 7.EE.B.4

Let the two equal sides be

x

and the third side be

y

.

2x + y = 12 \;\Rightarrow\; y = 12 - 2x

Once

x

is chosen, the perimeter decides

y

.

2The triangle condition gives a lower bound

#13 Convert to Algebra 7.G.A.2

The two equal sides together must be longer than the third side. (The other checks,

x + y > x

, hold automatically because

y > 0

.)

2x > 12 - 2x \;\Rightarrow\; 4x > 12 \;\Rightarrow\; x > 3

If the equal sides were too short, they could not reach each other over the base.

3A positive third side gives an upper bound

#13 Convert to Algebra 7.EE.B.4

The third side has to be a positive length.

12 - 2x > 0 \;\Rightarrow\; x < 6

The two equal sides cannot use up the whole perimeter.

4List the whole numbers in between

#2 Make a Systematic List 7.G.A.2

The whole numbers with

3 < x < 6

are

4, 5

. Each gives

y = 12 - 2x

.

(4, 4, 4), \quad (5, 5, 2)

Two triangles, including the equilateral one, which has all three sides equal.

Answer:

2

4 · Reviewdoes it hold up?

Each listed triple adds to $12$ and passes the triangle test: $5 + 2 > 5$ , and $4, 4, 4$ is equilateral. Just outside the range, $(3, 3, 6)$ fails because $3 + 3 = 6$ , which is not longer than $6$ .

Another way: List every whole $x$ from $1$ to $5$ with $y = 12 - 2x$ : $(1,1,10), (2,2,8), (3,3,6), (4,4,4), (5,5,2)$ . Cross out the first three, where the equal sides add to no more than the third side. Two remain.

Standardsmin grade 7

7.EE.B.4 Use variables to represent quantities and construct simple equations and inequalities — Writing the third side as $12 - 2x$ and turning the conditions into inequalities in $x$.
7.G.A.2 Draw geometric shapes with given conditions including triangles — Using the condition on three lengths that determine a triangle.

💡Takeaway. The triangle rule gives a lower bound, the perimeter gives an upper bound -- count the whole numbers between them.

Variant 2 easy

Among triangles whose three side lengths add up to $14$ , how many isosceles triangles (at least two sides equal) have all three side lengths whole numbers?

Your answer

Show solution

1 · Understandwhat's really being asked

Count the isosceles triangles with whole-number side lengths and perimeter $14$ .

Givens

The three side lengths add up to $14$ .
The triangle is isosceles: at least two sides are equal.
Every side length is a whole number.

Unknowns

How many such triangles there are.

Constraints

Three lengths make a triangle only if any two of them together are longer than the third.

2 · Planchoose the strategy

#13 Convert to Algebra · also uses: #2 Make a Systematic List

Name the equal sides $x$ and the third side $y$ . The perimeter turns $y$ into an expression in $x$ , the triangle condition gives a lower bound on $x$ , and the third side being positive gives an upper bound. Then list the whole numbers in between.

3 · Execute4 carry out the plan

1Write the third side in terms of the equal sides

#13 Convert to Algebra 7.EE.B.4

Let the two equal sides be

x

and the third side be

y

.

2x + y = 14 \;\Rightarrow\; y = 14 - 2x

Once

x

is chosen, the perimeter decides

y

.

2The triangle condition gives a lower bound

#13 Convert to Algebra 7.G.A.2

The two equal sides together must be longer than the third side. (The other checks,

x + y > x

, hold automatically because

y > 0

.)

2x > 14 - 2x \;\Rightarrow\; 4x > 14 \;\Rightarrow\; x > 3.5

If the equal sides were too short, they could not reach each other over the base.

3A positive third side gives an upper bound

#13 Convert to Algebra 7.EE.B.4

The third side has to be a positive length.

14 - 2x > 0 \;\Rightarrow\; x < 7

The two equal sides cannot use up the whole perimeter.

4List the whole numbers in between

#2 Make a Systematic List 7.G.A.2

The whole numbers with

3.5 < x < 7

are

4, 5, 6

. Each gives

y = 14 - 2x

.

(4, 4, 6), \quad (5, 5, 4), \quad (6, 6, 2)

Three triangles; none is equilateral, because

14

is not a multiple of

3

.

Answer:

3

4 · Reviewdoes it hold up?

Each listed triple adds to $14$ and passes the triangle test: $4 + 4 > 6$ , $5 + 4 > 5$ , and $6 + 2 > 6$ . Just outside the range, $(3, 3, 8)$ fails because $3 + 3 < 8$ .

Another way: List every whole $x$ from $1$ to $6$ with $y = 14 - 2x$ : $(1,1,12), (2,2,10), (3,3,8), (4,4,6), (5,5,4), (6,6,2)$ . Cross out the first three, where the equal sides add to less than the third side. Three remain.

Standardsmin grade 7

7.EE.B.4 Use variables to represent quantities and construct simple equations and inequalities — Writing the third side as $14 - 2x$ and turning the conditions into inequalities in $x$.
7.G.A.2 Draw geometric shapes with given conditions including triangles — Using the condition on three lengths that determine a triangle.

💡Takeaway. The triangle rule gives a lower bound, the perimeter gives an upper bound -- count the whole numbers between them.

Variant 3 easy

Among triangles whose three side lengths add up to $15$ , how many isosceles triangles (at least two sides equal) have all three side lengths whole numbers?

Your answer

Show solution

1 · Understandwhat's really being asked

Count the isosceles triangles with whole-number side lengths and perimeter $15$ .

Givens

The three side lengths add up to $15$ .
The triangle is isosceles: at least two sides are equal.
Every side length is a whole number.

Unknowns

How many such triangles there are.

Constraints

Three lengths make a triangle only if any two of them together are longer than the third.

2 · Planchoose the strategy

#13 Convert to Algebra · also uses: #2 Make a Systematic List

Name the equal sides $x$ and the third side $y$ . The perimeter turns $y$ into an expression in $x$ , the triangle condition gives a lower bound on $x$ , and the third side being positive gives an upper bound. Then list the whole numbers in between.

3 · Execute4 carry out the plan

1Write the third side in terms of the equal sides

#13 Convert to Algebra 7.EE.B.4

Let the two equal sides be

x

and the third side be

y

.

2x + y = 15 \;\Rightarrow\; y = 15 - 2x

Once

x

is chosen, the perimeter decides

y

.

2The triangle condition gives a lower bound

#13 Convert to Algebra 7.G.A.2

The two equal sides together must be longer than the third side. (The other checks,

x + y > x

, hold automatically because

y > 0

.)

2x > 15 - 2x \;\Rightarrow\; 4x > 15 \;\Rightarrow\; x > 3.75

If the equal sides were too short, they could not reach each other over the base.

3A positive third side gives an upper bound

#13 Convert to Algebra 7.EE.B.4

The third side has to be a positive length.

15 - 2x > 0 \;\Rightarrow\; x < 7.5

The two equal sides cannot use up the whole perimeter.

4List the whole numbers in between

#2 Make a Systematic List 7.G.A.2

The whole numbers with

3.75 < x < 7.5

are

4, 5, 6, 7

. Each gives

y = 15 - 2x

.

(4, 4, 7), \quad (5, 5, 5), \quad (6, 6, 3), \quad (7, 7, 1)

Four triangles, including the equilateral one, which has all three sides equal.

Answer:

4

4 · Reviewdoes it hold up?

Each listed triple adds to $15$ and passes the triangle test: $4 + 4 > 7$ , $6 + 3 > 6$ , $7 + 1 > 7$ , and $5, 5, 5$ is equilateral. Just outside the range, $(3, 3, 9)$ fails because $3 + 3 < 9$ .

Another way: List every whole $x$ from $1$ to $7$ with $y = 15 - 2x$ : $(1,1,13), (2,2,11), (3,3,9), (4,4,7), (5,5,5), (6,6,3), (7,7,1)$ . Cross out the first three, where the equal sides add to less than the third side. Four remain.

Standardsmin grade 7

7.EE.B.4 Use variables to represent quantities and construct simple equations and inequalities — Writing the third side as $15 - 2x$ and turning the conditions into inequalities in $x$.
7.G.A.2 Draw geometric shapes with given conditions including triangles — Using the condition on three lengths that determine a triangle.

💡Takeaway. The triangle rule gives a lower bound, the perimeter gives an upper bound -- count the whole numbers between them.

Variant 4 easy

Among triangles whose three side lengths add up to $16$ , how many isosceles triangles (at least two sides equal) have all three side lengths whole numbers?

Your answer

Show solution

1 · Understandwhat's really being asked

Count the isosceles triangles with whole-number side lengths and perimeter $16$ .

Givens

The three side lengths add up to $16$ .
The triangle is isosceles: at least two sides are equal.
Every side length is a whole number.

Unknowns

How many such triangles there are.

Constraints

Three lengths make a triangle only if any two of them together are longer than the third.

2 · Planchoose the strategy

#13 Convert to Algebra · also uses: #2 Make a Systematic List

Name the equal sides $x$ and the third side $y$ . The perimeter turns $y$ into an expression in $x$ , the triangle condition gives a lower bound on $x$ , and the third side being positive gives an upper bound. Then list the whole numbers in between.

3 · Execute4 carry out the plan

1Write the third side in terms of the equal sides

#13 Convert to Algebra 7.EE.B.4

Let the two equal sides be

x

and the third side be

y

.

2x + y = 16 \;\Rightarrow\; y = 16 - 2x

Once

x

is chosen, the perimeter decides

y

.

2The triangle condition gives a lower bound

#13 Convert to Algebra 7.G.A.2

The two equal sides together must be longer than the third side. (The other checks,

x + y > x

, hold automatically because

y > 0

.)

2x > 16 - 2x \;\Rightarrow\; 4x > 16 \;\Rightarrow\; x > 4

If the equal sides were too short, they could not reach each other over the base.

3A positive third side gives an upper bound

#13 Convert to Algebra 7.EE.B.4

The third side has to be a positive length.

16 - 2x > 0 \;\Rightarrow\; x < 8

The two equal sides cannot use up the whole perimeter.

4List the whole numbers in between

#2 Make a Systematic List 7.G.A.2

The whole numbers with

4 < x < 8

are

5, 6, 7

. Each gives

y = 16 - 2x

.

(5, 5, 6), \quad (6, 6, 4), \quad (7, 7, 2)

Three triangles; none is equilateral, because

16

is not a multiple of

3

.

Answer:

3

4 · Reviewdoes it hold up?

Each listed triple adds to $16$ and passes the triangle test: $5 + 5 > 6$ , $6 + 4 > 6$ , and $7 + 2 > 7$ . Just outside the range, $(4, 4, 8)$ fails because $4 + 4 = 8$ , which is not longer than $8$ .

Another way: List every whole $x$ from $1$ to $7$ with $y = 16 - 2x$ : $(1,1,14), (2,2,12), (3,3,10), (4,4,8), (5,5,6), (6,6,4), (7,7,2)$ . Cross out the first four, where the equal sides add to no more than the third side. Three remain.

Standardsmin grade 7

7.EE.B.4 Use variables to represent quantities and construct simple equations and inequalities — Writing the third side as $16 - 2x$ and turning the conditions into inequalities in $x$.
7.G.A.2 Draw geometric shapes with given conditions including triangles — Using the condition on three lengths that determine a triangle.

💡Takeaway. The triangle rule gives a lower bound, the perimeter gives an upper bound -- count the whole numbers between them.

Variant 5 medium

Among triangles whose three side lengths add up to $17$ , how many isosceles triangles (at least two sides equal) have all three side lengths whole numbers?

Your answer

Show solution

1 · Understandwhat's really being asked

Count the isosceles triangles with whole-number side lengths and perimeter $17$ .

Givens

The three side lengths add up to $17$ .
The triangle is isosceles: at least two sides are equal.
Every side length is a whole number.

Unknowns

How many such triangles there are.

Constraints

Three lengths make a triangle only if any two of them together are longer than the third.

2 · Planchoose the strategy

#13 Convert to Algebra · also uses: #2 Make a Systematic List

Name the equal sides $x$ and the third side $y$ . The perimeter turns $y$ into an expression in $x$ , the triangle condition gives a lower bound on $x$ , and the third side being positive gives an upper bound. Then list the whole numbers in between.

3 · Execute4 carry out the plan

1Write the third side in terms of the equal sides

#13 Convert to Algebra 7.EE.B.4

Let the two equal sides be

x

and the third side be

y

.

2x + y = 17 \;\Rightarrow\; y = 17 - 2x

Once

x

is chosen, the perimeter decides

y

.

2The triangle condition gives a lower bound

#13 Convert to Algebra 7.G.A.2

The two equal sides together must be longer than the third side. (The other checks,

x + y > x

, hold automatically because

y > 0

.)

2x > 17 - 2x \;\Rightarrow\; 4x > 17 \;\Rightarrow\; x > 4.25

If the equal sides were too short, they could not reach each other over the base.

3A positive third side gives an upper bound

#13 Convert to Algebra 7.EE.B.4

The third side has to be a positive length.

17 - 2x > 0 \;\Rightarrow\; x < 8.5

The two equal sides cannot use up the whole perimeter.

4List the whole numbers in between

#2 Make a Systematic List 7.G.A.2

The whole numbers with

4.25 < x < 8.5

are

5, 6, 7, 8

. Each gives

y = 17 - 2x

.

(5, 5, 7), \quad (6, 6, 5), \quad (7, 7, 3), \quad (8, 8, 1)

Four triangles; none is equilateral, because

17

is not a multiple of

3

.

Answer:

4

4 · Reviewdoes it hold up?

Each listed triple adds to $17$ and passes the triangle test: $5 + 5 > 7$ , $6 + 5 > 6$ , $7 + 3 > 7$ , and $8 + 1 > 8$ . Just outside the range, $(4, 4, 9)$ fails because $4 + 4 < 9$ .

Another way: List every whole $x$ from $1$ to $8$ with $y = 17 - 2x$ : $(1,1,15), (2,2,13), (3,3,11), (4,4,9), (5,5,7), (6,6,5), (7,7,3), (8,8,1)$ . Cross out the first four, where the equal sides add to less than the third side. Four remain.

Standardsmin grade 7

7.EE.B.4 Use variables to represent quantities and construct simple equations and inequalities — Writing the third side as $17 - 2x$ and turning the conditions into inequalities in $x$.
7.G.A.2 Draw geometric shapes with given conditions including triangles — Using the condition on three lengths that determine a triangle.

💡Takeaway. The triangle rule gives a lower bound, the perimeter gives an upper bound -- count the whole numbers between them.

Variant 6 medium

Among triangles whose three side lengths add up to $18$ , how many isosceles triangles (at least two sides equal) have all three side lengths whole numbers?

Your answer

Show solution

1 · Understandwhat's really being asked

Count the isosceles triangles with whole-number side lengths and perimeter $18$ .

Givens

The three side lengths add up to $18$ .
The triangle is isosceles: at least two sides are equal.
Every side length is a whole number.

Unknowns

How many such triangles there are.

Constraints

Three lengths make a triangle only if any two of them together are longer than the third.

2 · Planchoose the strategy

#13 Convert to Algebra · also uses: #2 Make a Systematic List

Name the equal sides $x$ and the third side $y$ . The perimeter turns $y$ into an expression in $x$ , the triangle condition gives a lower bound on $x$ , and the third side being positive gives an upper bound. Then list the whole numbers in between.

3 · Execute4 carry out the plan

1Write the third side in terms of the equal sides

#13 Convert to Algebra 7.EE.B.4

Let the two equal sides be

x

and the third side be

y

.

2x + y = 18 \;\Rightarrow\; y = 18 - 2x

Once

x

is chosen, the perimeter decides

y

.

2The triangle condition gives a lower bound

#13 Convert to Algebra 7.G.A.2

The two equal sides together must be longer than the third side. (The other checks,

x + y > x

, hold automatically because

y > 0

.)

2x > 18 - 2x \;\Rightarrow\; 4x > 18 \;\Rightarrow\; x > 4.5

If the equal sides were too short, they could not reach each other over the base.

3A positive third side gives an upper bound

#13 Convert to Algebra 7.EE.B.4

The third side has to be a positive length.

18 - 2x > 0 \;\Rightarrow\; x < 9

The two equal sides cannot use up the whole perimeter.

4List the whole numbers in between

#2 Make a Systematic List 7.G.A.2

The whole numbers with

4.5 < x < 9

are

5, 6, 7, 8

. Each gives

y = 18 - 2x

.

(5, 5, 8), \quad (6, 6, 6), \quad (7, 7, 4), \quad (8, 8, 2)

Four triangles, including the equilateral one, which has all three sides equal.

Answer:

4

4 · Reviewdoes it hold up?

Each listed triple adds to $18$ and passes the triangle test: $5 + 5 > 8$ , $7 + 4 > 7$ , $8 + 2 > 8$ , and $6, 6, 6$ is equilateral. Just outside the range, $(4, 4, 10)$ fails because $4 + 4 < 10$ .

Another way: List every whole $x$ from $1$ to $8$ with $y = 18 - 2x$ : $(1,1,16), (2,2,14), (3,3,12), (4,4,10), (5,5,8), (6,6,6), (7,7,4), (8,8,2)$ . Cross out the first four, where the equal sides add to less than the third side. Four remain.

Standardsmin grade 7

7.EE.B.4 Use variables to represent quantities and construct simple equations and inequalities — Writing the third side as $18 - 2x$ and turning the conditions into inequalities in $x$.
7.G.A.2 Draw geometric shapes with given conditions including triangles — Using the condition on three lengths that determine a triangle.

💡Takeaway. The triangle rule gives a lower bound, the perimeter gives an upper bound -- count the whole numbers between them.

Variant 7 medium

Among triangles whose three side lengths add up to $19$ , how many isosceles triangles (at least two sides equal) have all three side lengths whole numbers?

Your answer

Show solution

1 · Understandwhat's really being asked

Count the isosceles triangles with whole-number side lengths and perimeter $19$ .

Givens

The three side lengths add up to $19$ .
The triangle is isosceles: at least two sides are equal.
Every side length is a whole number.

Unknowns

How many such triangles there are.

Constraints

Three lengths make a triangle only if any two of them together are longer than the third.

2 · Planchoose the strategy

#13 Convert to Algebra · also uses: #2 Make a Systematic List

Name the equal sides $x$ and the third side $y$ . The perimeter turns $y$ into an expression in $x$ , the triangle condition gives a lower bound on $x$ , and the third side being positive gives an upper bound. Then list the whole numbers in between.

3 · Execute4 carry out the plan

1Write the third side in terms of the equal sides

#13 Convert to Algebra 7.EE.B.4

Let the two equal sides be

x

and the third side be

y

.

2x + y = 19 \;\Rightarrow\; y = 19 - 2x

Once

x

is chosen, the perimeter decides

y

.

2The triangle condition gives a lower bound

#13 Convert to Algebra 7.G.A.2

The two equal sides together must be longer than the third side. (The other checks,

x + y > x

, hold automatically because

y > 0

.)

2x > 19 - 2x \;\Rightarrow\; 4x > 19 \;\Rightarrow\; x > 4.75

If the equal sides were too short, they could not reach each other over the base.

3A positive third side gives an upper bound

#13 Convert to Algebra 7.EE.B.4

The third side has to be a positive length.

19 - 2x > 0 \;\Rightarrow\; x < 9.5

The two equal sides cannot use up the whole perimeter.

4List the whole numbers in between

#2 Make a Systematic List 7.G.A.2

The whole numbers with

4.75 < x < 9.5

are

5, 6, 7, 8, 9

. Each gives

y = 19 - 2x

.

(5, 5, 9), \quad (6, 6, 7), \quad (7, 7, 5), \quad (8, 8, 3), \quad (9, 9, 1)

Five triangles; none is equilateral, because

19

is not a multiple of

3

.

Answer:

5

4 · Reviewdoes it hold up?

Each listed triple adds to $19$ and passes the triangle test: $5 + 5 > 9$ , $6 + 6 > 7$ , $7 + 5 > 7$ , $8 + 3 > 8$ , and $9 + 1 > 9$ . Just outside the range, $(4, 4, 11)$ fails because $4 + 4 < 11$ .

Another way: List every whole $x$ from $1$ to $9$ with $y = 19 - 2x$ : $(1,1,17), (2,2,15), (3,3,13), (4,4,11), (5,5,9), (6,6,7), (7,7,5), (8,8,3), (9,9,1)$ . Cross out the first four, where the equal sides add to less than the third side. Five remain.

Standardsmin grade 7

7.EE.B.4 Use variables to represent quantities and construct simple equations and inequalities — Writing the third side as $19 - 2x$ and turning the conditions into inequalities in $x$.
7.G.A.2 Draw geometric shapes with given conditions including triangles — Using the condition on three lengths that determine a triangle.

💡Takeaway. The triangle rule gives a lower bound, the perimeter gives an upper bound -- count the whole numbers between them.

Variant 8 medium

Among triangles whose three side lengths add up to $20$ , how many isosceles triangles (at least two sides equal) have all three side lengths whole numbers?

Your answer

Show solution

1 · Understandwhat's really being asked

Count the isosceles triangles with whole-number side lengths and perimeter $20$ .

Givens

The three side lengths add up to $20$ .
The triangle is isosceles: at least two sides are equal.
Every side length is a whole number.

Unknowns

How many such triangles there are.

Constraints

Three lengths make a triangle only if any two of them together are longer than the third.

2 · Planchoose the strategy

#13 Convert to Algebra · also uses: #2 Make a Systematic List

Name the equal sides $x$ and the third side $y$ . The perimeter turns $y$ into an expression in $x$ , the triangle condition gives a lower bound on $x$ , and the third side being positive gives an upper bound. Then list the whole numbers in between.

3 · Execute4 carry out the plan

1Write the third side in terms of the equal sides

#13 Convert to Algebra 7.EE.B.4

Let the two equal sides be

x

and the third side be

y

.

2x + y = 20 \;\Rightarrow\; y = 20 - 2x

Once

x

is chosen, the perimeter decides

y

.

2The triangle condition gives a lower bound

#13 Convert to Algebra 7.G.A.2

The two equal sides together must be longer than the third side. (The other checks,

x + y > x

, hold automatically because

y > 0

.)

2x > 20 - 2x \;\Rightarrow\; 4x > 20 \;\Rightarrow\; x > 5

If the equal sides were too short, they could not reach each other over the base.

3A positive third side gives an upper bound

#13 Convert to Algebra 7.EE.B.4

The third side has to be a positive length.

20 - 2x > 0 \;\Rightarrow\; x < 10

The two equal sides cannot use up the whole perimeter.

4List the whole numbers in between

#2 Make a Systematic List 7.G.A.2

The whole numbers with

5 < x < 10

are

6, 7, 8, 9

. Each gives

y = 20 - 2x

.

(6, 6, 8), \quad (7, 7, 6), \quad (8, 8, 4), \quad (9, 9, 2)

Four triangles; none is equilateral, because

20

is not a multiple of

3

.

Answer:

4

4 · Reviewdoes it hold up?

Each listed triple adds to $20$ and passes the triangle test: $6 + 6 > 8$ , $7 + 6 > 7$ , $8 + 4 > 8$ , and $9 + 2 > 9$ . Just outside the range, $(5, 5, 10)$ fails because $5 + 5 = 10$ , which is not longer than $10$ .

Another way: List every whole $x$ from $1$ to $9$ with $y = 20 - 2x$ : $(1,1,18), (2,2,16), (3,3,14), (4,4,12), (5,5,10), (6,6,8), (7,7,6), (8,8,4), (9,9,2)$ . Cross out the first five, where the equal sides add to no more than the third side. Four remain.

Standardsmin grade 7

7.EE.B.4 Use variables to represent quantities and construct simple equations and inequalities — Writing the third side as $20 - 2x$ and turning the conditions into inequalities in $x$.
7.G.A.2 Draw geometric shapes with given conditions including triangles — Using the condition on three lengths that determine a triangle.

💡Takeaway. The triangle rule gives a lower bound, the perimeter gives an upper bound -- count the whole numbers between them.

Variant 9 hard

Among triangles whose three side lengths add up to $21$ , how many isosceles triangles (at least two sides equal) have all three side lengths whole numbers?

Your answer

Show solution

1 · Understandwhat's really being asked

Count the isosceles triangles with whole-number side lengths and perimeter $21$ .

Givens

The three side lengths add up to $21$ .
The triangle is isosceles: at least two sides are equal.
Every side length is a whole number.

Unknowns

How many such triangles there are.

Constraints

Three lengths make a triangle only if any two of them together are longer than the third.

2 · Planchoose the strategy

#13 Convert to Algebra · also uses: #2 Make a Systematic List

Name the equal sides $x$ and the third side $y$ . The perimeter turns $y$ into an expression in $x$ , the triangle condition gives a lower bound on $x$ , and the third side being positive gives an upper bound. Then list the whole numbers in between.

3 · Execute4 carry out the plan

1Write the third side in terms of the equal sides

#13 Convert to Algebra 7.EE.B.4

Let the two equal sides be

x

and the third side be

y

.

2x + y = 21 \;\Rightarrow\; y = 21 - 2x

Once

x

is chosen, the perimeter decides

y

.

2The triangle condition gives a lower bound

#13 Convert to Algebra 7.G.A.2

The two equal sides together must be longer than the third side. (The other checks,

x + y > x

, hold automatically because

y > 0

.)

2x > 21 - 2x \;\Rightarrow\; 4x > 21 \;\Rightarrow\; x > 5.25

If the equal sides were too short, they could not reach each other over the base.

3A positive third side gives an upper bound

#13 Convert to Algebra 7.EE.B.4

The third side has to be a positive length.

21 - 2x > 0 \;\Rightarrow\; x < 10.5

The two equal sides cannot use up the whole perimeter.

4List the whole numbers in between

#2 Make a Systematic List 7.G.A.2

The whole numbers with

5.25 < x < 10.5

are

6, 7, 8, 9, 10

. Each gives

y = 21 - 2x

.

(6, 6, 9), \quad (7, 7, 7), \quad (8, 8, 5), \quad (9, 9, 3), \quad (10, 10, 1)

Five triangles, including the equilateral one, which has all three sides equal.

Answer:

5

4 · Reviewdoes it hold up?

Each listed triple adds to $21$ and passes the triangle test: $6 + 6 > 9$ , $8 + 5 > 8$ , $9 + 3 > 9$ , $10 + 1 > 10$ , and $7, 7, 7$ is equilateral. Just outside the range, $(5, 5, 11)$ fails because $5 + 5 < 11$ .

Another way: List every whole $x$ from $1$ to $10$ with $y = 21 - 2x$ : $(1,1,19), (2,2,17), (3,3,15), (4,4,13), (5,5,11), (6,6,9), (7,7,7), (8,8,5), (9,9,3), (10,10,1)$ . Cross out the first five, where the equal sides add to less than the third side. Five remain.

Standardsmin grade 7

7.EE.B.4 Use variables to represent quantities and construct simple equations and inequalities — Writing the third side as $21 - 2x$ and turning the conditions into inequalities in $x$.
7.G.A.2 Draw geometric shapes with given conditions including triangles — Using the condition on three lengths that determine a triangle.

💡Takeaway. The triangle rule gives a lower bound, the perimeter gives an upper bound -- count the whole numbers between them.

Variant 10 hard

Among triangles whose three side lengths add up to $22$ , how many isosceles triangles (at least two sides equal) have all three side lengths whole numbers?

Your answer

Show solution

1 · Understandwhat's really being asked

Count the isosceles triangles with whole-number side lengths and perimeter $22$ .

Givens

The three side lengths add up to $22$ .
The triangle is isosceles: at least two sides are equal.
Every side length is a whole number.

Unknowns

How many such triangles there are.

Constraints

Three lengths make a triangle only if any two of them together are longer than the third.

2 · Planchoose the strategy

#13 Convert to Algebra · also uses: #2 Make a Systematic List

Name the equal sides $x$ and the third side $y$ . The perimeter turns $y$ into an expression in $x$ , the triangle condition gives a lower bound on $x$ , and the third side being positive gives an upper bound. Then list the whole numbers in between.

3 · Execute4 carry out the plan

1Write the third side in terms of the equal sides

#13 Convert to Algebra 7.EE.B.4

Let the two equal sides be

x

and the third side be

y

.

2x + y = 22 \;\Rightarrow\; y = 22 - 2x

Once

x

is chosen, the perimeter decides

y

.

2The triangle condition gives a lower bound

#13 Convert to Algebra 7.G.A.2

The two equal sides together must be longer than the third side. (The other checks,

x + y > x

, hold automatically because

y > 0

.)

2x > 22 - 2x \;\Rightarrow\; 4x > 22 \;\Rightarrow\; x > 5.5

If the equal sides were too short, they could not reach each other over the base.

3A positive third side gives an upper bound

#13 Convert to Algebra 7.EE.B.4

The third side has to be a positive length.

22 - 2x > 0 \;\Rightarrow\; x < 11

The two equal sides cannot use up the whole perimeter.

4List the whole numbers in between

#2 Make a Systematic List 7.G.A.2

The whole numbers with

5.5 < x < 11

are

6, 7, 8, 9, 10

. Each gives

y = 22 - 2x

.

(6, 6, 10), \quad (7, 7, 8), \quad (8, 8, 6), \quad (9, 9, 4), \quad (10, 10, 2)

Five triangles; none is equilateral, because

22

is not a multiple of

3

.

Answer:

5

4 · Reviewdoes it hold up?

Each listed triple adds to $22$ and passes the triangle test: $6 + 6 > 10$ , $7 + 7 > 8$ , $8 + 6 > 8$ , $9 + 4 > 9$ , and $10 + 2 > 10$ . Just outside the range, $(5, 5, 12)$ fails because $5 + 5 < 12$ .

Another way: List every whole $x$ from $1$ to $10$ with $y = 22 - 2x$ : $(1,1,20), (2,2,18), (3,3,16), (4,4,14), (5,5,12), (6,6,10), (7,7,8), (8,8,6), (9,9,4), (10,10,2)$ . Cross out the first five, where the equal sides add to less than the third side. Five remain.

Standardsmin grade 7

7.EE.B.4 Use variables to represent quantities and construct simple equations and inequalities — Writing the third side as $22 - 2x$ and turning the conditions into inequalities in $x$.
7.G.A.2 Draw geometric shapes with given conditions including triangles — Using the condition on three lengths that determine a triangle.

💡Takeaway. The triangle rule gives a lower bound, the perimeter gives an upper bound -- count the whole numbers between them.

Variant 11 hard

Among triangles whose three side lengths add up to $24$ , how many isosceles triangles (at least two sides equal) have all three side lengths whole numbers?

Your answer

Show solution

1 · Understandwhat's really being asked

Count the isosceles triangles with whole-number side lengths and perimeter $24$ .

Givens

The three side lengths add up to $24$ .
The triangle is isosceles: at least two sides are equal.
Every side length is a whole number.

Unknowns

How many such triangles there are.

Constraints

Three lengths make a triangle only if any two of them together are longer than the third.

2 · Planchoose the strategy

#13 Convert to Algebra · also uses: #2 Make a Systematic List

Name the equal sides $x$ and the third side $y$ . The perimeter turns $y$ into an expression in $x$ , the triangle condition gives a lower bound on $x$ , and the third side being positive gives an upper bound. Then list the whole numbers in between.

3 · Execute4 carry out the plan

1Write the third side in terms of the equal sides

#13 Convert to Algebra 7.EE.B.4

Let the two equal sides be

x

and the third side be

y

.

2x + y = 24 \;\Rightarrow\; y = 24 - 2x

Once

x

is chosen, the perimeter decides

y

.

2The triangle condition gives a lower bound

#13 Convert to Algebra 7.G.A.2

The two equal sides together must be longer than the third side. (The other checks,

x + y > x

, hold automatically because

y > 0

.)

2x > 24 - 2x \;\Rightarrow\; 4x > 24 \;\Rightarrow\; x > 6

If the equal sides were too short, they could not reach each other over the base.

3A positive third side gives an upper bound

#13 Convert to Algebra 7.EE.B.4

The third side has to be a positive length.

24 - 2x > 0 \;\Rightarrow\; x < 12

The two equal sides cannot use up the whole perimeter.

4List the whole numbers in between

#2 Make a Systematic List 7.G.A.2

The whole numbers with

6 < x < 12

are

7, 8, 9, 10, 11

. Each gives

y = 24 - 2x

.

(7, 7, 10), \quad (8, 8, 8), \quad (9, 9, 6), \quad (10, 10, 4), \quad (11, 11, 2)

Five triangles, including the equilateral one, which has all three sides equal.

Answer:

5

4 · Reviewdoes it hold up?

Each listed triple adds to $24$ and passes the triangle test: $7 + 7 > 10$ , $9 + 6 > 9$ , $10 + 4 > 10$ , $11 + 2 > 11$ , and $8, 8, 8$ is equilateral. Just outside the range, $(6, 6, 12)$ fails because $6 + 6 = 12$ , which is not longer than $12$ .

Another way: List every whole $x$ from $1$ to $11$ with $y = 24 - 2x$ : $(1,1,22), (2,2,20), (3,3,18), (4,4,16), (5,5,14), (6,6,12), (7,7,10), (8,8,8), (9,9,6), (10,10,4), (11,11,2)$ . Cross out the first six, where the equal sides add to no more than the third side. Five remain.

Standardsmin grade 7

7.EE.B.4 Use variables to represent quantities and construct simple equations and inequalities — Writing the third side as $24 - 2x$ and turning the conditions into inequalities in $x$.
7.G.A.2 Draw geometric shapes with given conditions including triangles — Using the condition on three lengths that determine a triangle.

💡Takeaway. The triangle rule gives a lower bound, the perimeter gives an upper bound -- count the whole numbers between them.

Variant 12 hard

Among triangles whose three side lengths add up to $27$ , how many isosceles triangles (at least two sides equal) have all three side lengths whole numbers?

Your answer

Show solution

1 · Understandwhat's really being asked

Count the isosceles triangles with whole-number side lengths and perimeter $27$ .

Givens

The three side lengths add up to $27$ .
The triangle is isosceles: at least two sides are equal.
Every side length is a whole number.

Unknowns

How many such triangles there are.

Constraints

Three lengths make a triangle only if any two of them together are longer than the third.

2 · Planchoose the strategy

#13 Convert to Algebra · also uses: #2 Make a Systematic List

Name the equal sides $x$ and the third side $y$ . The perimeter turns $y$ into an expression in $x$ , the triangle condition gives a lower bound on $x$ , and the third side being positive gives an upper bound. Then list the whole numbers in between.

3 · Execute4 carry out the plan

1Write the third side in terms of the equal sides

#13 Convert to Algebra 7.EE.B.4

Let the two equal sides be

x

and the third side be

y

.

2x + y = 27 \;\Rightarrow\; y = 27 - 2x

Once

x

is chosen, the perimeter decides

y

.

2The triangle condition gives a lower bound

#13 Convert to Algebra 7.G.A.2

The two equal sides together must be longer than the third side. (The other checks,

x + y > x

, hold automatically because

y > 0

.)

2x > 27 - 2x \;\Rightarrow\; 4x > 27 \;\Rightarrow\; x > 6.75

If the equal sides were too short, they could not reach each other over the base.

3A positive third side gives an upper bound

#13 Convert to Algebra 7.EE.B.4

The third side has to be a positive length.

27 - 2x > 0 \;\Rightarrow\; x < 13.5

The two equal sides cannot use up the whole perimeter.

4List the whole numbers in between

#2 Make a Systematic List 7.G.A.2

The whole numbers with

6.75 < x < 13.5

are

7, 8, 9, 10, 11, 12, 13

. Each gives

y = 27 - 2x

.

(7, 7, 13), \quad (8, 8, 11), \quad (9, 9, 9), \quad (10, 10, 7), \quad (11, 11, 5), \quad (12, 12, 3), \quad (13, 13, 1)

Seven triangles, including the equilateral one, which has all three sides equal.

Answer:

7

4 · Reviewdoes it hold up?

Each listed triple adds to $27$ and passes the triangle test: $7 + 7 > 13$ , $8 + 8 > 11$ , $10 + 7 > 10$ , $11 + 5 > 11$ , $12 + 3 > 12$ , $13 + 1 > 13$ , and $9, 9, 9$ is equilateral. Just outside the range, $(6, 6, 15)$ fails because $6 + 6 < 15$ .

Another way: List every whole $x$ from $1$ to $13$ with $y = 27 - 2x$ : $(1,1,25), (2,2,23), (3,3,21), (4,4,19), (5,5,17), (6,6,15), (7,7,13), (8,8,11), (9,9,9), (10,10,7), (11,11,5), (12,12,3), (13,13,1)$ . Cross out the first six, where the equal sides add to less than the third side. Seven remain.

Standardsmin grade 7

7.EE.B.4 Use variables to represent quantities and construct simple equations and inequalities — Writing the third side as $27 - 2x$ and turning the conditions into inequalities in $x$.
7.G.A.2 Draw geometric shapes with given conditions including triangles — Using the condition on three lengths that determine a triangle.

💡Takeaway. The triangle rule gives a lower bound, the perimeter gives an upper bound -- count the whole numbers between them.

Triangle inequality bounds isosceles legs · 12 practice problems

Generated variants — 12

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