Draw a radius from the center of the circle to the outer end of the segment. What can be said about the triangle formed?
7
Practice makes perfect
Let's take a look at the given diagram. We are asked to find the value of x.
Let's draw a radius from the center of the circle to the outer end of the segment with a length of 7. Keep in mind that since the radius is constant, its length is always 6. As a result, we get an isosceles triangle.
We will find the measure of the vertex angle in the other triangle by finding the measure of the arc that creates it. We are given two arc angles in the prompt, the 65^(∘) that we already noted and 230^(∘) around the rest of the circle. Since the measure of a full turn around a circle is 360^(∘), we can use the Arc Addition Postulate to find the missing measure.
360^(∘)-230^(∘)-65^(∘)= 65^(∘)
We have found that the vertex angle of the left triangle measures 65^(∘).
Note that in both triangles, the vertex angles measure 65^(∘) and the legs are the same length, 6. By the Side-Angle-Side Congurence Theorem, these two triangles are congruent. Because corresponding elements of congruent figures are by definition congruent, the bases of these triangles must also be congruent.
x= 7
