Question Video: Finding the Measure of an Angle Using the Properties of Tangents to the Circle | Nagwa Question Video: Finding the Measure of an Angle Using the Properties of Tangents to the Circle | Nagwa

# Question Video: Finding the Measure of an Angle Using the Properties of Tangents to the Circle Mathematics • Third Year of Preparatory School

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Given that π΄π· is a tangent to the circle and πβ π·π΄πΆ = 90Β°, calculate πβ π΄πΆπ΅.

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### Video Transcript

Given that π΄π· is a tangent to the circle and the measure of angle π·π΄πΆ is 90 degrees, calculate the measure of angle π΄πΆπ΅.

We are told in the question that π΄π· is a tangent and the measure of angle π·π΄πΆ is 90 degrees. We know that the tangent to any circle is perpendicular to the radius or diameter. This means that, in this question, π΄πΆ is a diameter of the circle. We could use two possible angle properties or circle theorems to solve this problem. Firstly, we could use the fact that the angle in a semicircle equals 90 degrees. This means that the measure of angle π΄π΅πΆ is 90 degrees. We could also have found this using the alternate segment theorem, where the measure of angle π·π΄πΆ is equal to the measure of angle π΄π΅πΆ. Either way, we know that π΄π΅πΆ is equal to 90 degrees.

We now need to solve the equation nine π₯ is equal to 90. Dividing both sides of this equation by nine gives us π₯ is equal to 10. Our value of π₯ is 10 degrees. We can see on the diagram that the measure of angle π΄πΆπ΅ is five π₯. As π₯ is equal to 10 degrees, we need to multiply this by five. Five multiplied by 10 is equal to 50. Therefore, angle π΄πΆπ΅ equals 50 degrees.

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