Depth · one concept from first principles · step 2 of 4
Triangle inequality bounds isosceles legs
Count the isosceles triangles with whole-number side lengths and perimeter 18.
GeometryExpressions
Your answer
How to solve
Strategy Convert to Algebra — 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.
1STEP 1
Write the third side in terms of the equal sides
Let the two equal sides be x and the third side be y.
2x + y = 18 → y = 18 - 2x
Once x is chosen, the perimeter decides y.
7.EE.B.4Convert To Algebra2STEP 2
The triangle condition gives a lower bound
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 → 4x ＞ 18 → x ＞ 4.5
If the equal sides were too short, they could not reach each other over the base.
7.G.A.2Convert To AlgebraThe two equal sides together must be longer than the third side, so .
Why?
Going from one end of the third side to the other by way of the top corner is never shorter than going straight, and if the two equal sides only matched the third side, the triangle would flatten into a line.
🧱Two sides must together beat the thirdGoing from one corner to another by way of a third is never shorter than going straight, so two sides laid end to end must exceed the third or the triangle flattens.
Why?
Adding to both sides of slides both the same distance along the number line, so the order stays and .
🧱Adding to both sides keeps an inequalitySliding both numbers the same distance along the number line keeps the one on the right on the right, so adding or subtracting the same amount keeps an inequality's direction.
3STEP 3
A positive third side gives an upper bound
The third side has to be a positive length.
18 - 2x ＞ 0 → x ＜ 9
The two equal sides cannot use up the whole perimeter.
7.EE.B.4Convert To Algebra4STEP 4
List the whole numbers in between
The whole numbers with 4.5 ＜ x ＜ 9 are 5, 6, 7, 8. Each gives y = 18 - 2x.
(5, 5, 8), (6, 6, 6), (7, 7, 4), (8, 8, 2)
Four triangles, including the equilateral one, which has all three sides equal.
7.G.A.2Make A Systematic ListAnswer
4
All four listed triangles have side lengths adding to 18 and pass 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.
Takeaway
The triangle rule gives a lower bound, the perimeter gives an upper bound -- count the whole numbers between them.
- Write the third side in terms of the equal sides
- The triangle condition gives a lower bound
- A positive third side gives an upper bound
- List the whole numbers in between
Where next?
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