Question

Construct any two similar triangles with the ratio of corresponding sides 7:3 and prove that the triangles are similar.

Answer 1

1. Draw any triangle with base BC .
2. From B, draw a ray BX. Cut 7 arcs of equal length on the ray BX at $B_1,B_2,..B_7$.
3. Draw a line from $B_7$ to C.
4. From $B_3$, draw another line parallel to $B_{7}C$, cutting the line BC at $C_1$.
5. From $C_1$, draw another line parallel to AC, cutting the line BA at $A_1$.
Triangle $A_{1}B{C}_1$ is the required similar triangle to ABC with the ratio of corresponding sides 7 :3.

Proof:
Since $A_1{C}_1$ || AC,
Angles $A_1{C}_{1}B$ and ACB are equal (corresponding angles).
Similarly, angles $B{A}_1{C}_1$ and BAC will also be equal.
Hence by AA similarity, we can say that triangles ABC and $A_{1}B{C}_1$ are similar.
So the ratio of corresponding sides are equal and are equal to 7:3 by construction.