Question

Construct a triangle with sides 4 cm, 5 cm, and 6 cm and then construct another triangle similar to it whose sides are \( \frac{2}{3} \) of the corresponding sides of the first triangle.

Hint:

Use the Basic Proportionality Theorem (BPT) to construct a similar triangle with a given scale factor.

Answer 1

Step 1: Constructing the original triangle
1. Draw a base \( BC = 6 \) cm.
2. Using a compass, draw an arc of radius 4 cm from \( B \).
3. Draw another arc of radius 5 cm from \( C \), intersecting the first arc at \( A \).
4. Connect \( A \) to \( B \) and \( C \) to form \( \triangle ABC \).
Step 2: Constructing the similar triangle
1. Draw a ray \( BX \) making an acute angle with \( BC \).
2. Mark 3 equal divisions along \( BX \) (since the required ratio is \( 2:3 \)).
3. Connect the third point to \( C \).
4. Draw a line parallel to \( C_3C \) through the second division point, meeting \( BC \) at \( C' \).
5. Draw a line parallel to \( AC \) through \( C' \), meeting \( AB \) at \( A' \).
Step 3: Triangle Verification
The triangle \( \triangle A'B'C' \) is similar to \( \triangle ABC \) by Basic Proportionality Theorem (BPT).
Thus, the required triangle with sides \( \frac{2}{3} \) of the original is successfully constructed.