How to Find the Area of a Triangle Inside a Rectangle

Geometry math problem: Find the area of the triangle

The figure shows a quarter circle a circle which is tangent to each other. AB and BC are tangents of the circle and DC is the tangent of both circle and quarter circle. Find the area of the blue triangle when the radius of the circle is 1 cm and the radius of the quarter circle is 2 cm

Solution

From the above figure, we can write Area of blue Triangle = Area of triangle AOP

Extent PO then we get a triangle AOQ with the same height as triangle AOP

⇒ Area of blue triangle = Area of triangle AOP

Apply Pythagoras theorem in triangle AOQ

AQ² = AO² − OQ²

⇒ AQ² = 3² − 1² = 9 − 1 = 8

⇒ AQ = 2√2 cm

Area of triangle AOP = ½ × AQ × OQ = ½ × 2√2 × 1

⇒ Area of triangle AOP = √2 cm²

⇒ Area of blue triangle = √2 cm²