The length of one of the sides of an acute angled triangle is 13 units. If the area of the triangle is 90 units^2 and the length of the another side of the triangle is 15 units. Find the length of the third side.
A. √124
B. √134
C. √224
D. √234
E. √244
OA E
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The length of one of the sides of an acute angled triangle
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Take the side of length 15 as the base. If we draw a height from that base, then since the area is 90, that height must be 12.
When we draw that height, we divide the triangle into two smaller right triangles, each of which has, as its hypotenuse, one of the other two sides of the big triangle. So one of those two right triangles has a hypotenuse of 13, and one of its sides is 12, so it's a 5-12-13 triangle. That '5' is part of our base of 15, so the other of our two smaller right triangles has two sides of length 10 (the rest of the base) and 12 (the height of the triangle), and using Pythagoras we can find the missing side, which is what the question asks for:
10^2 + 12^2 = d^2
244 = d^2
√244 = d
That root can (and must) be simplified, by taking out the √4 from √244, so I don't like how they've written the answer choices.
When we draw that height, we divide the triangle into two smaller right triangles, each of which has, as its hypotenuse, one of the other two sides of the big triangle. So one of those two right triangles has a hypotenuse of 13, and one of its sides is 12, so it's a 5-12-13 triangle. That '5' is part of our base of 15, so the other of our two smaller right triangles has two sides of length 10 (the rest of the base) and 12 (the height of the triangle), and using Pythagoras we can find the missing side, which is what the question asks for:
10^2 + 12^2 = d^2
244 = d^2
√244 = d
That root can (and must) be simplified, by taking out the √4 from √244, so I don't like how they've written the answer choices.
For online GMAT math tutoring, or to buy my higher-level Quant books and problem sets, contact me at ianstewartgmat at gmail.com
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