Q) If circles are drawn taking two sides of a triangle as diameters, prove that the point of intersection of these circles lies on the third side.

9th Class Maths – NCERT Important Questions

Ans:

Step 1:

Let’s make a diagram for better understanding of the question:

Here, in Δ PQR, 2 circles are drawn – Circle 1 (pink) on PQ as diameter and Circle 2 (blue) on PR as diameter.

These circles intersect each other at points P and T.

We need to prove that point T lies on side QR.

Step 2:

Let’s connect points P and T. 

Now in circle 1, PQ is the diameter

∴  ∠ PTQ = 90 ⁰

Similarly, in circle 2, PR is the diameter

∴  ∠ PTR = 90 ⁰

Step 3:

Now in Δ PQR, ∠ PTQ + ∠ PTR = 90 ⁰ + 90 ⁰ = 180 ⁰

Thus, these 2 angles form a linear pair

But ∵ a linear pair of angles can exist only on a straight line

∴ QR is a straight line and point T lies on side QR.

…. Hence Proved !

Therefore, the point of intersection of the circles lies on the third side.

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