Diagonals of a trapezium PQRS intersect each other at the point O, PQ||RS and PQ = 3 RS. Find the ratio of the areas of triangles POQ and ROS
Solution:
Given, the diagonals of a trapezium PQRS intersect at point O
Also, PQ||RS and PQ = 3RS
We have to find the ratio of the areas of triangles POQ and ROS.
A trapezium is a two-dimensional quadrilateral having a pair of parallel opposite sides. The opposite parallel sides are referred to as the base and the non-parallel sides are referred to as legs of the trapezium.
Since, PQ||RS and PQ = 3RS
PQ/RS = 3/1 = 3 -------------- (1)
In △POQ and △ROS,
The vertically opposite angles ∠SOR and ∠POQ are equal.
i.e, ∠SOR = ∠POQ
The alternate angles ∠SRP and ∠QPR are equal.
i.e., ∠SRP = ∠QPR
AAA criterion states that if two angles of a triangle are respectively equal to two angles of another triangle, then by the angle sum property of a triangle their third angle will also be equal.
By AAA criterion, the third angle will be equal.
i.e., ∠RSO = ∠OQP
Therefore, the triangles POQ and ROS are similar.
By the property of an area of similar triangles,
Area of the triangle POQ/Area of the triangle ROS = PQ²/RS²
Substituting (1) in the above relation,
Area of the triangle POQ/Area of the triangle ROS = 3²/1²
= 9/1
Therefore, the ratio of the area of the triangles POQ and ROS is 9:1
✦ Try This: In a square PQRS, diagonals bisect each other at O. Prove that: ∆ POQ = ∆ QOR = ∆ ROS= ∆ SOP.
☛ Also Check: NCERT Solutions for Class 10 Maths Chapter 6
NCERT Exemplar Class 10 Maths Exercise 6.3 Problem 4
Diagonals of a trapezium PQRS intersect each other at the point O, PQ||RS and PQ = 3 RS. Find the ratio of the areas of triangles POQ and ROS
Summary:
Diagonals of a trapezium PQRS intersect each other at the point O, PQ||RS and PQ = 3 RS. The ratio of the areas of triangles POQ and ROS is 9:1
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