Explore How should the rectangle be constructed so that the diagonal divides the opposite angles into equal parts?

Question

Homi · Accepted Answer

Understand the Question

Each corner angle of a rectangle measures $90^\circ$ .
If a diagonal divides an angle into two equal parts, each part must measure $\dfrac{90^\circ}{2} = 45^\circ$ .
In the resulting triangle formed by the diagonal and two adjacent sides, having two $45^\circ$ angles makes it an isosceles right-angled triangle, meaning the adjacent sides must be equal.
A rectangle with equal adjacent sides is a square.

Step 1 · Determine the Divided Angles

Consider a rectangle $ABCD$ with diagonal $AC$ . Diagram 1

Each interior angle of a rectangle is $90^\circ$ .

Since diagonal $AC$ divides the opposite angles into equal parts: $\text{Each part} = \dfrac{90^\circ}{2} = 45^\circ$

Therefore $\angle BAC = \angle DAC = 45^\circ$ $\angle BCA = \angle DCA = 45^\circ$

Step 2 · Compare the Sides of the Rectangle

In $\Delta ABC$ , $\angle B = 90^\circ$ and $\angle BAC = 45^\circ$ .

\begin{aligned} \angle BCA &= 180^\circ - \angle B - \angle BAC \\ &= 180^\circ - 90^\circ - 45^\circ \\ &= 90^\circ - 45^\circ \\ &= 45^\circ \end{aligned}

Since $\angle BAC = \angle BCA = 45^\circ$ , the sides opposite to equal angles are equal: $AB = BC$

A rectangle with equal adjacent sides is a square.

Answer

The rectangle must be constructed as a square (with all sides equal).

Common Mistakes

Assuming all rectangles bisect angles: In a general rectangle, diagonals do not divide corner angles into equal halves; this only occurs when adjacent sides are equal.
Overlooking side equality: Forgetting that equal base angles ( $45^\circ$ ) in $\Delta ABC$ make it an isosceles triangle, requiring adjacent sides $AB = BC$ .

Question 7