AMC 8 2005 Problem 23
AMC8 2005 年第 23 题
三角形★★★☆☆
Isosceles right triangle ABC encloses a semicircle of area . The circle has its center O on hypotenuse and is tangent to sides and . What is the area of triangle ABC ?
等腰直角三角形 内含一个面积为 的半圆。该半圆圆心 在斜边 上，且与边 和 相切。那么三角形 的面积是多少？
- A.6
- B.8
- C.
- D.10
- E.
正确答案 · Correct Answer
B
解析 · Solution
Solution 1
First, we notice half a square so first let's create a square. Once we have a square, we will have a full circle. This circle has a diameter of 4 which will be the side of the square. The area would be Divide 16 by 2 to get the original shape and you get 8
We can figure out the radius of the semicircle from the question states that the area of the semicircle is and we can multiply it by 2 to complete the circle to get which we can find the radius of 2. Draw line segment OD such that it is the midsegment of triangle ABC, using the midsegment theorem we can see that line segment AC = 2*2=4. Since triangle ABC is an isosceles right triangle we can calculate the area to be = 8
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