Square diagonals are equal and perpendicular bisectors

4.G.A.24.MD.A.3 · adapt · grade 4

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

The figure shows a square with side length $16\,\text{cm}$ containing an inscribed circle. Four points on the circle are connected to form a second, inner square $ABCD$ . Find the length, in centimeters, of segment $BM$ .

[Figure] A large square with side length $16\,\text{cm}$ has a circle inscribed in it, touching all four sides. The four points where the circle meets the sides (the midpoints of the large square's sides) are joined to form the inner square $ABCD$ . The two diagonals $AC$ and $BD$ of the inner square meet at the center point $M$ .

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Understand

A big square has side 16 cm with a circle drawn inside touching all four sides. The four touch points (the midpoints of the big square's sides) are joined to make an inner square ABCD. Its diagonals AC and BD cross at the center M. I need the length of BM.

Givens

The outer square has side length 16 cm.
The inscribed circle touches all four sides at the midpoints of the outer square's sides.
Those four midpoints are A (top), B (right), C (bottom), D (left), forming inner square ABCD.
Diagonals AC and BD of the inner square meet at center M.

Unknowns

The length of segment BM.

Constraints

A (top midpoint) and C (bottom midpoint) lie on the vertical center line, so AC spans the full height of the outer square.
B (right midpoint) and D (left midpoint) lie on the horizontal center line, so BD spans the full width of the outer square.
A square's diagonals are equal and bisect each other at M.

Plan

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

Read the picture carefully: the inner square's diagonals are just the center lines of the outer square, so each diagonal equals the outer side. Then halve it because the diagonals bisect each other.

Execute

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

B is the midpoint of the right side and D is the midpoint of the left side. The segment BD runs straight across the middle of the outer square, so its length equals the outer square's side, 16 cm.

BD = 16 \text{ cm}

Connecting opposite side-midpoints of a square gives a segment as long as the side itself.

#7 Identify Subproblems 4.G.A.2

The diagonals of the inner square bisect each other at center M, so M is the midpoint of BD. Therefore BM = BD / 2 = 16 / 2 = 8 cm.

16 \div 2 = 8 \text{ cm}

A square's diagonals cut each other exactly in half, so BM is half of 16 cm.

Answer: 8 cm

Review

BM = 8 cm is half the 16 cm width, which matches that M is the exact center of the figure (also the circle's center, radius 8 cm). The size and centimeter unit are sensible.

Subproblem view (tool 7): the diagonal BD equals the outer side and also equals the circle's diameter (2 x radius = 2 x 8 = 16 cm); half of that diameter is the radius 8 cm = BM, confirming the answer.

Standards · min grade 4

4.G.A.2 Classify two-dimensional figures based on presence of parallel or perpendicular lines — Using square diagonal properties (equal, perpendicular, bisecting) and that midpoint-to-midpoint equals a side.
4.MD.A.3 Apply area and perimeter formulas for rectangles in real-world problems — Reasoning with the square's side length to get the diagonal and half-diagonal lengths.

💡 Joining opposite side-midpoints of a square gives a line as long as the side, and the diagonals split in half -- so BM is just 16 divided by 2!