GRE Math Challenge Question #28 :
What is the area of an equilateral triangle with vertices at (-1, -3), (9, -3), and (m, n) where m and n are both positive numbers?A. 25√2
B. 50√2
C. 10√3
D. 25√3
E. 50√3
Explanation:
To find the area of an equilateral triangle with vertices at (-1, -3), (9, -3), and (m, n), you do not need to find the values of m and n. To find the area of an equilateral triangle, you only need one side. So, you should first find the distance between (-1, -3) and (9, -3).
Since these two points are on a horizontal line together (they share a y-coordinate), the distance is just the difference between their x-coordinates: 9 - (-1) = 10.
An equilateral triangle with side 10 will have the same area regardless of where it is placed on an xy-coordinate plane, so the location of m and n is irrelevant. Instead, draw an equilateral triangle with sides equal 10. Drop a height down the middle. 10
Dividing a 60–60–60 triangle in this way creates two 30–60–90 triangles. The bottom side of the triangle is bisected by the height:
Using the properties of 30–60–90 triangles, h is equal to the shortest side multiplied by the square root of 3. Thus, b = 5√3 (You may also wish to memorize that the height of an equilateral triangle is always equal to half the side multiplied by √3.)
Find the area of the triangle, using 10 as the base:
A = bh/2
= (5√3)*10 / 2
= 25√3
Correct Choice : (D)
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