Video: CBSE Class X • Pack 2 • 2017 • Question 7

CBSE Class X • Pack 2 • 2017 • Question 7

03:03

Video Transcript

Prove that the tangents drawn at the endpoints of a chord of a circle form equal angles with the chord.

Here is a circle with a cord 𝑋𝑌. We’ll call the center of the circle 𝑂. Here’s the tangent of 𝑋 and the tangent of 𝑌. And we’re trying to prove that these two angles are equal. To do that, let’s label the intersection of the tangents 𝑇. This means we’re trying to prove angle 𝑇𝑋𝑌 is equal to angle 𝑇𝑌𝑋.

First, we say 𝑋𝑌 is a chord. Line 𝑂𝑋 is a radius of the circle. Line 𝑂𝑌 is also a radius. And that means 𝑂𝑋 is equal to 𝑂𝑌. The next thing we can say is that the angle 𝑂𝑌𝑇, the angle created by a radius and the tangent at a point, equals 90 degrees. This is also true of angle 𝑂𝑋𝑇. The intersection of a line of a radius and its tangent at a point is perpendicular, 90 degrees.

Now, we want to connect the center of the circle to the intersection of the two tangents. This will create two right triangles, triangle 𝑂𝑌𝑇 and triangle 𝑂𝑋𝑇. Both of these triangles have a right angle. They are right triangles. They have two side lengths that are equal. And they share a hypotenuse. They have a hypotenuse of equal length. This means that triangle 𝑂𝑌𝑇 is congruent to triangle 𝑂𝑋𝑇. This is proven by Right Hypotenuse Side Congruence.

Since we know that these two triangles are congruent, we can say that side length 𝑋𝑇 is equal to side length 𝑌𝑇. Remember what we’re trying to prove. We’re trying to prove that the angles made with the chord are equal. Our original chord is 𝑋𝑌. And we know that angles opposite equal sides are equal. Angle 𝑋𝑌𝑇 is opposite side 𝑋𝑇. And angle 𝑌𝑋𝑇 is opposite side 𝑌𝑇. Therefore, angle 𝑇𝑋𝑌 is equal to angle 𝑇𝑌𝑋.

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