Question 7

If a diagonal cuts a rectangle's angles in half, the rectangle must have equal sides.

Step 1 — Understanding the angles

Let us draw a rectangle. Let us call its corners A, B, C, and D. Let us draw a diagonal from corner A to corner C. A rectangle has four corner angles. Each corner angle is 90 degrees. The problem says the diagonal divides the opposite angles into equal parts. This means the diagonal AC cuts angle A into two equal parts. It also means the diagonal AC cuts angle C into two equal parts. So, angle BAC must be equal to angle DAC. And angle BCA must be equal to angle DCA. Each part is half of 90 degrees.

$\text{Each part} = \frac{90^\circ}{2}$

$\boxed{45^\circ}$

So, angle BAC is $45^\circ$. Angle DAC is $45^\circ$. Angle BCA is $45^\circ$. Angle DCA is $45^\circ$.

Step 2 — Looking at a triangle

Let us look at the triangle ABC. This triangle has three angles. Angle B is a corner of the rectangle. So, angle B is 90 degrees. From Step 1, we know angle BAC is 45 degrees. The sum of angles in any triangle is 180 degrees. So, we can find angle BCA.

$\text{Angle BCA} = 180^\circ - \text{Angle B} - \text{Angle BAC}$

$\text{Angle BCA} = 180^\circ - 90^\circ - 45^\circ$

$\text{Angle BCA} = 90^\circ - 45^\circ$

$\boxed{45^\circ}$

We see that angle BAC is $45^\circ$ and angle BCA is $45^\circ$. In triangle ABC, two angles are equal. When two angles in a triangle are equal, the sides opposite them are also equal. The side opposite angle BCA is AB. The side opposite angle BAC is BC. So, side AB must be equal to side BC. This means the length of side AB is the same as the length of side BC. A rectangle with all its sides equal is called a square.

Answer

The rectangle must be a square.