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Question 8:
Question Stem:
In the figure above, the four vertices of square RSTU lie on circle O, which has a radius of 8. What is the area of the shaded region?
Options:
(A) 32
(B) 96
(C) 128
(D) 150
(E) 166
Chart/Diagram Description:
Type: Geometric figure.
Main Elements:
- A circle with center labeled O.
- A square labeled RSTU is inscribed in the circle. The vertices R, S, T, and U are on the circle. The vertices are labeled in counter-clockwise order, with R at the top left and S at the top right.
- The center O is inside the square.
- There is a shaded region within the square. The shaded region appears to be the area of the square RSTU excluding the area of the triangle ROS.
- The shaded region is bounded by the sides UT, TU, a portion of the side UR, a portion of the side TS, and lines from O to points on UR and TS (specifically, the points R and S). Visually, the shaded region is the part of the square below the line segment RS, where RS is a side of the triangle ROS.
- Point O is connected by lines to R and S, forming triangle ROS, which is unshaded.
Other Relevant Text:
The radius of circle O is 8.
RSTU is a square.
The four vertices of square RSTU lie on circle O.
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视频信息
答案文本
视频字幕
In this problem, we have a circle with center O and radius 8. A square RSTU is inscribed in the circle, meaning all four vertices of the square lie on the circle. We need to find the area of the shaded region, which is the area of the square minus the area of triangle ROS. Let's start by understanding the relationship between the square and the circle.
To find the area of the shaded region, we first need to determine the side length of the square. Since the square is inscribed in the circle, its diagonal is equal to the diameter of the circle. The diameter is twice the radius, so the diameter equals 16. The diagonal of a square with side length s is s times the square root of 2. Setting these equal, we get s times the square root of 2 equals 16. Solving for s, we get s equals 16 divided by the square root of 2, which simplifies to 8 times the square root of 2.
Now that we know the side length of the square is 8 times the square root of 2, we can calculate its area. The area of a square is the square of its side length. So the area equals 8 times the square root of 2, squared. This equals 64 times 2, which gives us 128 square units. This is the total area of the square RSTU. In the next step, we'll calculate the area of the triangle ROS, which we need to subtract to find the shaded region.
Now we need to find the area of triangle ROS, which is the unshaded portion of the square. Since O is the center of the circle, OR and OS are both radii of the circle, so they both equal 8. The angle ROS is 90 degrees because the square is inscribed in the circle, and the central angle for each side of the square is 90 degrees. Using the formula for the area of a triangle, which is one-half times the base times the height, we get one-half times 8 times 8, which equals 32 square units. This is the area of the unshaded triangle ROS.