sat suite question viewer

Geometry and Trigonometry / Circles Difficulty: Hard

A circle has center $O$ , and points $R$ and $S$ lie on the circle. In triangle $O R S$ , the measure of $∠ R O S$ is $88 degree$ . What is the measure of $∠ R S O$ , in degrees? (Disregard the degree symbol when entering your answer.)

Back question 12 of 49 Next

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49

Explanation

The correct answer is $46$ . It's given that $O$ is the center of a circle and that points $R$ and $S$ lie on the circle. Therefore, $\bar{O R}$ and $\bar{O S}$ are radii of the circle. It follows that $O R = O S$ . If two sides of a triangle are congruent, then the angles opposite them are congruent. It follows that the angles $∠ R S O$ and $∠ O R S$ , which are across from the sides of equal length, are congruent. Let $x °$ represent the measure of $∠ R S O$ . It follows that the measure of $∠ O R S$ is also $x °$ . It's given that the measure of $∠ R O S$ is $88 degree$ . Because the sum of the measures of the interior angles of a triangle is $180 degree$ , the equation $x ° + x ° + 88 ° = 180 °$ , or $2 x + 88 = 180$ , can be used to find the measure of $∠ R S O$ . Subtracting $88$ from both sides of this equation yields $2 x equals 92$ . Dividing both sides of this equation by $2$ yields $x equals 46$ . Therefore, the measure of $∠ R S O$ , in degrees, is $46$ .