Question

Given that AB is a tangent of the circle with the center at X, AB = 12, and XD = 2.5, which is the length of DB?

Answer 1

Answer on Question#38094 - Math - Geometry

Question: Given that $AB$ is a tangent of the circle with the center at $X$ , $AB = 12$ , and $XD = 2.5$ , which is the length of DB?

Solution. Let us first make a drawing:

Recall that by definition of a tangent of a circle, $AB$ is perpendicular to the radius $XA$ . Thus, the triangle $ABD$ is a right triangle, and we can use the Pythagorean theorem:

$$A D ^ {2} + A B ^ {2} = D B ^ {2}.$$

Note that $AD$ is the diameter of our circle, and $AD = 2XD$ .

We have

$$D B ^ {2} = (2 X D) ^ {2} + A B ^ {2} = 5 ^ {2} + 1 2 ^ {2} = 1 6 9,$$

and thus

$$D B = \sqrt {1 6 9} = 1 3.$$

Answer. $DB = 13$