AMC 8 2015 Problem 6

AMC8 2015 年第 6 题

三角形★★☆☆☆
In △ABC\bigtriangleup ABC , AB=BC=29 , and AC=42 . What is the area of △ABC\bigtriangleup ABC ?
在 △ABC\triangle ABC 中，AB=BC=29，AC=42。那么 △ABC\triangle ABC 的面积是多少？
  1. A.100
  2. B.420
  3. C.500
  4. D.609
  5. E.701

正确答案 · Correct Answer

B

解析 · Solution

Solution 1

We know the semi-perimeter of △ABC\triangle ABC is 29+29+422=50\displaystyle \frac{29+29+42}{2} = 50. Next, we use Heron's Formula to find that the area of the triangle is just 50(50−29)2(50−42)=50⋅212⋅8=420\sqrt{50(50-29)^2(50-42)}=\sqrt{50 \cdot 21^2 \cdot 8}=420. Splitting the isosceles triangle in half, we get a right triangle with hypotenuse 29 and leg 21. Using the Pythagorean Theorem, we know the height is 292−212=20\sqrt{29^2-21^2}=20. Now that we know the height, the area is (20)(42)2=\dfrac{(20)(42)}{2} =.

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