We have seen how to get 90° using paper folding. Now, suppose we do not have any paper but two sticks of equal length, and a thread. How do we make an exact 90° using these?

Question

Homi · Accepted Answer

We can use the properties of congruent triangles to make a 90-degree angle.

Step 1 — Setting up the sticks

Let us take two sticks. Let their length be $L$ . We will call them Stick AB and Stick CD. We find the midpoint of each stick. Let M be the midpoint of Stick AB. So, AM = MB = $L/2$ . Let M also be the midpoint of Stick CD. So, CM = MD = $L/2$ . We place the sticks so their midpoints meet. We fix them at this common point M. This allows the sticks to rotate around M.

Diagram 1

Step 2 — Using the thread

We use the thread to measure distances. We measure the distance from point A to point D. Let us call this distance $d_1$ . We then measure the distance from point B to point D. Let us call this distance $d_2$ . We slowly move the sticks around M. We adjust their angle until $d_1$ and $d_2$ are equal. So, we make sure that AD = BD. Once AD = BD, we tighten the screw at M. This fixes the sticks in their position.

Diagram 2

Step 3 — Proving congruence

Now, let us look at two triangles. Consider triangle AMD ( $\triangle AMD$ ). Consider triangle BMD ( $\triangle BMD$ ). We know that AM = BM. This is because M is the midpoint of AB. We also know that AD = BD. We made these distances equal using the thread. The side MD is common to both triangles. So, MD = MD. We have three pairs of equal sides. This is the SSS (Side-Side-Side) congruence condition. Therefore, $\triangle AMD$ and $\triangle BMD$ are congruent.

$AM = BM \quad (\text{M is midpoint of AB})$ $AD = BD \quad (\text{by construction})$ $MD = MD \quad (\text{common side})$

$\boxed{\triangle AMD \cong \triangle BMD \quad (\text{by SSS rule})}$

Step 4 — Finding the angle

Since the triangles are congruent, their corresponding angles are equal. So, angle AMD ( $\angle AMD$ ) equals angle BMD ( $\angle BMD$ ). Points A, M, and B lie on a straight line. So, angles $\angle AMD$ and $\angle BMD$ form a linear pair. The sum of angles in a linear pair is 180 degrees. So, $\angle AMD + \angle BMD = 180^\circ$ . Since $\angle AMD = \angle BMD$ , we can substitute. $\angle AMD + \angle AMD = 180^\circ$ $2 \times \angle AMD = 180^\circ$ $\angle AMD = \frac{180^\circ}{2}$

$\boxed{\angle AMD = 90^\circ}$ Therefore, the angle between the sticks is 90 degrees.

Answer

The angle between the sticks is 90 degrees.

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