Chapter 9, Problem 8CR

To determine

To obtain: the measure of XY

Answer to Problem 8CR

The measure of XY is $13\sqrt{2}$

Explanation of Solution

Given Information: A circle with two tangents intersecting the circle at two points and measure of angle of intersection of the tangents being 90⁰. The length of one of the tangents is 13.

Formula used:

(a) Tangent drawn to a circle from a common point to intersect the circle at two different points has the same measure.

  

It can be expressed as,

  $\overline{PA}\cong \overline{PB}$

(b) The relationship between the legs of right triangle (shown in figure below) a , b and its hypotenuse c is obtained by using Pythagoras theorem.

  

It can be expressed as,

  $\begin{array}{l}{c}^2=a^2+b^2\\ c=\sqrt{a^2+b^2}\end{array}$

Calculation:

Consider the circle centered at P with tangents XZ and ZY as shown below. The $\angle Z$ is given as 90⁰. Here, this makes the triangle XYZ a right triangle with XY being the hypotenuse.

  

Now, length of one of the tangents XZ is given as XZ=13. By the property in (a) the length of the other tangent YZ =XZ=13. Thus, the ΔXYZ is an isosceles right triangle.

So applying Pythagoras theorem using (b) , in ΔXYZ, the length of XY can be obtained as,

  $\begin{array}{c}X{Y}^2=X{Z}^2+Y{Z}^2\\ XY=\sqrt{X{Z}^2+Y{Z}^2}\\ =\sqrt{{13}^2+{13}^2}\\ =13\sqrt{2}\end{array}$