Question Video: Finding the Measure of an Angle using the Relationship Between Tangents, Radii and Angles in a Semicircle | Nagwa Question Video: Finding the Measure of an Angle using the Relationship Between Tangents, Radii and Angles in a Semicircle | Nagwa

# Question Video: Finding the Measure of an Angle using the Relationship Between Tangents, Radii and Angles in a Semicircle Mathematics • Third Year of Preparatory School

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Given that π΅πΆ is a tangent to the circle with center π and πβ π΄ππ· = 97Β°, find πβ πΆπ΅π·.

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### Video Transcript

Given that π΅πΆ is a tangent to the circle with center π and the measure of angle π΄ππ· is 97 degrees, find the measure of angle πΆπ΅π·.

We know that in any circle, a tangent and radius or tangent and diameter meet at 90 degrees. We are told in the question that the measure of angle π΄ππ· is 97 degrees. We need to calculate the measure of the angle πΆπ΅π·. The alternate segment theorem states that, in any circle, the angle between a chord and a tangent through one of the endpoints of the chord is equal to the angle in the alternate segment. In this question, the measure of angle πΆπ΅π· is equal to angle π΅π΄π·.

Letβs now consider how we can calculate the angle π΅π΄π· using the information on the diagram. Triangle ππ΄π· is isosceles as the line segment ππ΄ is equal to the line segment ππ·. They are both radii of the circle. This means that the angles inside the triangle, angle ππ΄π· and angle ππ·π΄, are equal. These can be calculated by subtracting 97 from 180 and then dividing by two. 180 minus 97 is equal to 83. Dividing this by two or halving it gives us 41.5. Angles ππ΄π· and ππ·π΄ are both equal to 41.5 degrees. As the angle ππ΄π· is the same as the angle π΅π΄π·, we can see, using the alternate segment theorem, that the measure of angle πΆπ΅π· is also 41.5 degrees.

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