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20 Jun 2015, 11:12
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ABC is a right triangle with right angle at point B. If the ratio of AB to BC is 8 to 15, what is the area of the triangle?
(1) The length of the perpendicular from point B to the hypotenuse is 120.
(2) The perimeter of the triangle is 680.
(1) The length of the perpendicular from point B to the hypotenuse is 120.
(2) The perimeter of the triangle is 680.
20 Jun 2015, 11:12
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Official Solution:
ABC is a right triangle with right angle at point B. If the ratio of AB to BC is 8 to 15, what is the area of the triangle?
First of all, since the ration of AB to BC is \(8x:15x\), then the hypotenuse is \(\sqrt{(8x)^2+(15x)^2}=17x\).
(1) The length of the perpendicular from point B to the hypotenuse is 120.
Consider the image below:
The area of a right triangle equals to \(\frac{1}{2}*leg_1*leg_2=\frac{1}{2}**AB*BC\). But the area of a right triangle can also be found by \(\frac{1}{2}*(altitude \ from \ right \ angle)*(hypotenuse)=\frac{1}{2}*BD*AC\).
Equate these expressions:
\(\frac{1}{2}*AB*BC=\frac{1}{2}*BD*AC\);
\(\frac{1}{2}*8x*15x=\frac{1}{2}*120*17x\);
\(120x=120*17\);
\(x=17\).
The area = \(\frac{1}{2}*AB*BC=\frac{1}{2}*8x*15x=60x^2=60*17^2\). Sufficient.
(2) The perimeter of the triangle is 680.
The perimeter = \(8x+15x+17x=680\). We can find x, thus we can find the area. Sufficient.
Answer: D
ABC is a right triangle with right angle at point B. If the ratio of AB to BC is 8 to 15, what is the area of the triangle?
First of all, since the ration of AB to BC is \(8x:15x\), then the hypotenuse is \(\sqrt{(8x)^2+(15x)^2}=17x\).
(1) The length of the perpendicular from point B to the hypotenuse is 120.
Consider the image below:
The area of a right triangle equals to \(\frac{1}{2}*leg_1*leg_2=\frac{1}{2}**AB*BC\). But the area of a right triangle can also be found by \(\frac{1}{2}*(altitude \ from \ right \ angle)*(hypotenuse)=\frac{1}{2}*BD*AC\).
Equate these expressions:
\(\frac{1}{2}*AB*BC=\frac{1}{2}*BD*AC\);
\(\frac{1}{2}*8x*15x=\frac{1}{2}*120*17x\);
\(120x=120*17\);
\(x=17\).
The area = \(\frac{1}{2}*AB*BC=\frac{1}{2}*8x*15x=60x^2=60*17^2\). Sufficient.
(2) The perimeter of the triangle is 680.
The perimeter = \(8x+15x+17x=680\). We can find x, thus we can find the area. Sufficient.
Answer: D
04 Feb 2018, 12:39
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Hi,
a really stupid question, how do we get 17x out of this equation?
Thank you for your answer!
a really stupid question, how do we get 17x out of this equation?
Thank you for your answer!
