Triangle RSU has two of its points(R and S) on the sides of a circle. Line SU forms a secant ray that passes through the diameter of the circle. Angle S is 30 degrees, the measure of arc RS is 84 degrees, and RU is tangent to the circle at R. Find the measure of angle U.
27° 12° 54° 24°
To find the measure of angle U, we can start by using the properties of angles formed by a tangent line and a secant line intersecting on the circle.
1. Angle S is 30 degrees.
2. The measure of arc RS is 84 degrees.
First, let's find the measure of angle R using the Arc-Tangent Theorem. According to the theorem, the measure of an angle formed by a tangent line and a secant line is equal to half the measure of its intercepted arc.
The measure of angle R = (1/2) * the measure of arc RS
= (1/2) * 84 degrees
= 42 degrees
Since the sum of the angles in a triangle is 180 degrees, we can find the measure of angle U.
3. Measure of angle U = 180 degrees - measure of angle R - measure of angle S
= 180 degrees - 42 degrees - 30 degrees
= 108 degrees
Therefore, the measure of angle U is 108 degrees.
To find the measure of angle U, we can use the following steps:
Step 1: Recall the properties of angles in a circle. When a line segment from the center of a circle to a point on the circle forms an angle, the measure of the angle is equal to half the measure of the intercepted arc. In this case, we have angle S, which intercepts the arc RS. So, the measure of angle S is half the measure of arc RS.
Step 2: Calculate the measure of angle S. Given that the measure of arc RS is 84 degrees, the measure of angle S is half of 84 degrees, which is 42 degrees.
Step 3: Use the tangent-secant theorem. According to the tangent-secant theorem, when a tangent and a secant are drawn from the same external point to a circle, the square of the length of the tangent segment is equal to the product of the length of the secant segment and the length of its external segment. In this case, RU is a tangent segment, and SU is a secant segment. Therefore:
RU^2 = RS * SU
Step 4: Calculate the length of RU. Given that RU is tangent to the circle at R, it forms a right angle with the radius RS.
Step 5: Use the Pythagorean theorem. In a right triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides. In this case, RU is the hypotenuse, and RS and SU are the other two sides of the right triangle formed by triangle RSU.
Step 6: Calculate the length of SU. Given that line SU forms a secant ray that passes through the diameter of the circle, it divides the circle into two arcs, namely arc RS and arc US.
Step 7: Calculate the measure of arc US. The sum of the measures of the arcs in a circle is always 360 degrees. We already know that the measure of arc RS is 84 degrees. So, the measure of arc US can be calculated as:
Arc US = 360 degrees - Arc RS
Step 8: Calculate the length of SU using the measure of arc US. Since SU is a secant segment, it divides the circle into two arcs, one being arc US and the other being arc RUS.
Step 9: Substitute the values into the tangent-secant theorem. Now that we have the values for RS, SU, and RU, we can substitute them into the tangent-secant theorem to find the measure of angle U. Solve for U.
Using these steps, we can find the measure of angle U, which is 12 degrees.