Question

Prove that two triangles on the same base (or equal bases) and between the same parallels are equal in area.
[4 MARKS]

Answer 1

Diagram : 1 Mark
Concept : 1 Mark
Application of the concept: 1 Mark
Proof : 1 Mark

Let ABC and PBC are the 2 trangles on same base BC and between same parallels.

Drawing line segments DC and RC parallel to AB and BP respectively, we get 2 parallelograms ABCD and PBCR as shown below.

ar (ABCD) = ar (PBCR) -----(i) (Area of parallelograms on same bases and between same parallels are equal.)

Now ar ($△$ABC) = ar ($△$CDA) and ar ($△$PBC) = ar ( $△$CRP) (Diagonal divides parallelogram into 2 triangles of equal area)

So, ar ($△$ABC) = $\frac{1}{2}$ ar (ABCD)

and ar ($△$PBC) = $\frac{1}{2}$ ar (PBCR)

But ar (ABCD) = ar (PBCR) [ By (i)]

Therefore, ar ($△$ABC) = ar ($△$PBC)