Diagonals of a parallelogram bisect each other

4.G.A.2 · adapt · grade 4

Archetype: Quadrilateral Diagonal Properties · step in a 2-type progression

In parallelogram $ABCD$ , find the perimeter, in centimeters, of triangle $BCM$ .

[Figure] Parallelogram $ABCD$ has its top side running from $A$ (top-left) to $D$ (top-right) and its bottom side from $B$ (bottom-left) to $C$ (bottom-right). The two diagonals $AC$ and $BD$ meet at point $M$ . Side $AD$ measures $7\,\text{cm}$ and side $AB$ measures $5\,\text{cm}$ . The full diagonal $BD$ measures $6\,\text{cm}$ , and along diagonal $AC$ the segment $AM$ is labeled $5.4\,\text{cm}$ .

Show solution

Understand

In parallelogram ABCD the two diagonals AC and BD cross at M. I know AD = 7 cm, AB = 5 cm, the whole diagonal BD = 6 cm, and the half-diagonal AM = 5.4 cm. I need the perimeter of triangle BCM (the three sides BC, CM, MB added up).

Givens

ABCD is a parallelogram with diagonals AC and BD meeting at M.
AD = 7 cm and AB = 5 cm.
Full diagonal BD = 6 cm.
Half-diagonal AM = 5.4 cm.

Unknowns

The perimeter of triangle BCM, i.e. BC + CM + MB.

Constraints

The diagonals of a parallelogram bisect each other, so M is the midpoint of both BD and AC.
Opposite sides of a parallelogram are equal, so BC = AD.

Plan

#7 Identify Subproblems · also uses: #1 Draw a Diagram

Find the three sides of triangle BCM one at a time using two parallelogram facts: diagonals bisect each other (gives MB and CM) and opposite sides are equal (gives BC). Then add.

Execute

#7 Identify Subproblems 4.G.A.2

M is the midpoint of diagonal BD because a parallelogram's diagonals cut each other in half. So MB = BD / 2 = 6 / 2 = 3 cm.

6 \div 2 = 3 \text{ cm}

Diagonals of a parallelogram split each other evenly, so half of the 6 cm diagonal is 3 cm.

#7 Identify Subproblems 4.G.A.2

M is also the midpoint of diagonal AC, so CM equals the other half AM. Therefore CM = AM = 5.4 cm.

CM = AM = 5.4 \text{ cm}

Since M halves AC, the piece from M to C is the same 5.4 cm as the piece from A to M.

#1 Draw a Diagram 4.G.A.2

In a parallelogram opposite sides are equal, and BC is opposite AD, so BC = AD = 7 cm.

BC = AD = 7 \text{ cm}

Opposite sides of a parallelogram match, so BC copies AD's 7 cm.

#7 Identify Subproblems 4.MD.A.3

Perimeter of triangle BCM = MB + CM + BC = 3 + 5.4 + 7 = 15.4 cm.

3 + 5.4 + 7 = 15.4 \text{ cm}

Adding the three side lengths gives the distance all the way around the triangle.

Answer: 15.4 cm

Review

The three sides 3 cm, 5.4 cm, and 7 cm are each shorter than their sum and obey the triangle rule (3 + 5.4 = 8.4 > 7), so a real triangle exists. The total 15.4 cm is a sensible perimeter in centimeters.

Draw the diagram (tool 1) and mark the two equal halves on each diagonal; reading MB, MC, and BC straight off the marked figure gives the same 3, 5.4, and 7 cm.

Standards · min grade 4

4.G.A.2 Classify two-dimensional figures based on presence of parallel or perpendicular lines — Using parallelogram properties: diagonals bisect each other and opposite sides are equal.
4.MD.A.3 Apply area and perimeter formulas for rectangles in real-world problems — Adding the three side lengths to get the perimeter of triangle BCM.

💡 A parallelogram's diagonals chop each other exactly in half, so half-lengths plus a matching side give the perimeter with just Grade 4 addition!