# How to Find the Area of the Quadrilateral inside a Square

## Geometry math problem: Find the area of the quadrilateral inside a square

ABCD Is square also PA: PB = 3: 2, Connect BD and PC also measure OD = 25 cm Then find the area of quadrilateral APOD

## Solution

Blue area = Area of ∆ABD – Area of ∆POB

Let’s consider the side of square = 5k, then

PA = 3k

PB = 2k

Also consider, OB = x

Consider ∆OCD and ∆OPB

We know CD and PA are parallel lines

Then

CD/PB = 25/x

(5k)/(2k) = 25/x = 5/2

So, x = 50/5 = 10 cm

Also BD = 25 + 10 = 35* *cm

## Apply Pythagorean theorem in ∆ABD

Then BD = 5k√2 = 35

k = 7/√2 cm

Now draw heights of the triangle

From ∆ODC and ∆POB

(5k – h)/h = 5/2

h = 5√2 cm

So Blue area = ½ (5k)² – ½ 2kh

= ½ 25 (7/√2)² – 7/√2 × 5√2

= 1085 / 4 cm²

**Area of quadrilateral = 271.25 cm²**