Question

Please help me with the following problem:O is the center of the circumference. The lines XP and XQ are tangent to the circumference on points P and Q, respectively.Which is the measure of the amplitude of the angles a, b and c.

Answer 1

Step-by-step explanation: To solve this question we need to pay attention to some statements presented below

- The sum of the inner angles for any triangle is equal to 180°

- Once we have a line that is a tangent to a circumference there is a line that passes through the center of the circumference that is perpendicular to the first line.

Step 1: Let's take a look over the angles a. As we can see lines XP, PQ and QX build the triangle XPQ. We can also visualize that the lines PQ and XQ are similar which means the inner angles of the triangle XPQ are 70°, a° and a°. Now considering that the sum of the inner angles of a triangle is equal to 180° let's calculate angle a as follows

Step 2: Now looking over angle b we can see that the lines XQ and QO are perpendiculars which means there is a 90° angle between them. Now we can use this knowledge and the fact angle a = 55° to calculate angle b as follows

Step 3: Now looking over angle c we can see that there is another triangle inside the circle that seems to be a right triangle. Once the right triangle has a 90° angle we can assume that the inner angles of this triangle are 58°, 90° and c. Now we can calculate b as follows

Final answer: So the final answers are a = 55°, b = 35° and c = 32°.