The portal has been deactivated. Please contact your portal admin.

Question Video: Finding the Measure of an Angle Using the Properties of Triangles and Tangents to a Circle Mathematics

In the given figure, line 𝐴𝐡 is a tangent to a circle with center 𝑀 at 𝐡, line segment 𝐢𝐷 is a diameter, π‘šβˆ π΅π΄π‘€ = π‘₯, and π‘šβˆ π‘€π·π΅ = 2π‘₯ βˆ’ 55. Find the value of π‘₯ in degrees.

02:48

Video Transcript

In the given figure, line 𝐴𝐡 is a tangent to a circle with center 𝑀 at 𝐡, line segment 𝐢𝐷 is a diameter, the measure of angle 𝐡𝐴𝑀 equals π‘₯, and the measure of angle 𝑀𝐷𝐡 equals two π‘₯ minus 55. Find the value of π‘₯ in degrees.

We are told in the question that the line 𝐴𝐡 is a tangent to the circle at point 𝐡. We know that a tangent meets the radius at the point of contact at right angles. The measure of angle 𝐴𝐡𝑀 is therefore equal to 90 degrees. As all radii are equal in length, we know that the line segments 𝑀𝐷 and 𝑀𝐡 have the same length. This means that triangle 𝑀𝐷𝐡 is isosceles. This in turn means that the measures of angles 𝑀𝐷𝐡 and 𝑀𝐡𝐷 are equal. And they are equal to the expression two π‘₯ minus 55. We are also told in the question that the measure of angle 𝐡𝐴𝑀 is equal to π‘₯.

We will now consider triangle 𝐴𝐡𝐷 and the fact that angles in a triangle sum to 180 degrees. This means that the measures of the angles 𝐴𝐷𝐡, 𝐴𝐡𝐷, and 𝐡𝐴𝐷 must equal 180 degrees. Angle 𝐴𝐷𝐡 is equal to two π‘₯ minus 55, angle 𝐴𝐡𝐷 is equal to two π‘₯ minus 55 plus 90, and angle 𝐡𝐴𝐷 is equal to π‘₯.

This gives us the following equation. Two π‘₯ minus 55 plus two π‘₯ minus 55 plus 90 plus π‘₯ is equal to 180. Collecting like terms on the left-hand side gives us five π‘₯ minus 20, and this is equal to 180. We can then add 20 to both sides. Finally, dividing through by five gives us π‘₯ is equal to 40. The value of π‘₯ in degrees is 40 degrees.

Nagwa uses cookies to ensure you get the best experience on our website. Learn more about our Privacy Policy.