Question

ABCD and APDR are parallelograms. P is the mid-point of BC and Q is on CD such that DQ : QC = 2 : 1. Area (ΔPQC) = 20 cm². Find the areas of 

1. ΔPDC
2. ΔABP
3. || gm APDR

Answer 1

i. Area of ΔPDC

The area of ΔPQC is given as 20 cm².

Point Q is on CD such that DQ : QC = 2 : 1.

This means QC = `1/3` CD.

Triangles ΔPQC and ΔPDC share the same height from vertex P to the base CD.

The ratio of the areas of triangles with the same height is equal to the ratio of their bases.

Therefore, `("Area" (ΔPQC))/("Area" (ΔPDC)) = (QC)/(DC)`.

Substituting the known values, `20/("Area" (ΔPDC)) = 1/3`.

The area of ΔPDC is calculated as 20 × 3 = 60 cm².

ii. Area of ΔABP

P is the mid-point of BC, so BP = PC.

In parallelogram ABCD, AB = CD.

The area of ΔPDC is 60 cm².

The area of ΔBCD is twice the area of ΔPDC because P is the midpoint of BC and triangles ΔPDC and ΔPDB share the same height from D to BC and have equal bases PC and PB.

Thus, Area (ΔBCD) = 2 × Area (ΔPDC)

= 2 × 60

= 120 cm²

In a parallelogram, a diagonal divides it into two triangles of equal area. 

Therefore, Area(ΔBCD) = Area(ΔABD) = 120 cm².

The area of parallelogram ABCD is 2 × Area(ΔBCD) = 2 × 120 = 240 cm².

The area of ΔABP can be found by considering that P is the midpoint of BC.

The area of ΔABP is half the area of ΔABC because AP is a median to BC.

The area of ΔABC is equal to the area of ΔADC in parallelogram ABCD.

The area of ΔADC is equal to the area of ΔABD, which is 120 cm².

Therefore, Area(ΔABC) = 120 cm².

The area of ΔABP is `1/2` × Area(ΔABC) = `1/2` × 120 = 60 cm².

iii. Area of Parallelogram ΔPDR

APDR is a parallelogram.

The area of a parallelogram is given by base times height.

In parallelogram ABCD, AD is parallel to BC.

In parallelogram APDR, AD is parallel to PR.

The area of parallelogram APDR is equal to the area of parallelogram ABCD because they share the same base AD and the same height (perpendicular distance between AD and BC).

The area of parallelogram ABCD was found to be 240 cm².

Therefore, the area of parallelogram APDR is 240 cm².