Question

State true or false:

The ratio of the areas of two triangles on the same base is equal to the ratio of their heights.

• A. True
• B. False

Answer 1

The correct option is A True

To prove: Ratio of the areas of two triangles of the same bases is equal to the ratio of their heights.
Proof:
Let us take triangle $T_1$ with height $h_1$ and base $b_1$ and Triangle $T_2$ with height $h_2$ and base $b_2$.
Area of Triangle $T_1$ $=\frac{1}{2}\times \text{base}\times \text{height}=\frac{1}{2}\times b_1\times h_1$
Area of Triangle $T_2$ $=\frac{1}{2}\times \text{base}\times \text{height}=\frac{1}{2}\times b_2\times h_2$
Ratio of area of two triangles,
$\frac{\text{Area of triangle}{T}_1}{\text{Area of triangle}{T}_2}=\frac{\frac{1}{2}\times b_1\times h_1}{\frac{1}{2}\times b_2\times h_2}$
$=\frac{h_1}{h_2}$ (Base of two triangles are equal. So, $b_1=b_2$)
Proved.