Question

A triangle PQR is right angled at Q. E and F are mid points of QR and PR respectively. What will be the ratio of the area of the quadrilateral PQEF to the area of triangle PQR.
1. \(\frac{3}{4}\)
2. \(\frac{4}{3}\)
3. \(\frac{2}{3}\)
4. \(\frac{3}{2}\)

Answer 1

Correct Answer - Option 1 : \(\frac{3}{4}\)

Given :

A triangle PQR is right-angled at Q. E and F are midpoints of QR and PR respectively.

Concept used :

The area of a triangle is directly proportional to the square of the side of the triangle. 

In a triangle ABC if E and F are midpoints of side AB and AC then EF will be half of BC 

Calculation :

EF = (1/2) QP  (Concept given above)

Area of triangle REF/Area of triangle PQR = (EF/QP)²

⇒ 1/4 

Area of triangle REF : Area of PQR = 1 : 4 

Area of triangle = Area to triangle REF + Area of quad PQEF 

Let the area of 4 units 

Area of quad PQEF = 4 - 1 

⇒ 3 units 

So, 

Area of quad PQEF : Area of triangle PQR = 3 : 4 

∴ Option 1 will be the correct answer.