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A
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Difficulty:
55%
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Question Stats:
60% (01:42) correct
40%
(02:06)
wrong
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The area of the right triangle ABC is 4 times greater than the area of the right triangle KLM. If the hypotenuse KL is 10 inches, what is the length of the hypotenuse AB?
(1) Angles ABC and KLM are each equal to 55 degrees.
(2) LM is 6 inches.
I thought the answer was D because Statement 1 tells us that the triangles are similar and statement 2 tells us that KLM = a right 6-8-10 triangle. So the area of KLM is 24 which would make ABC’s area = 96 and which makes ABC = 12-16-20 triangle.
OPEN DISCUSSION OF THIS QUESTION IS HERE: the-area-of-the-right-triangle-abc-is-4-times-greater-than-127070.html
11 May 2010, 04:34
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TXTDryFly
The above solution is not right.
Properties of Similar Triangles:
• Corresponding angles are the same.
• Corresponding sides are all in the same proportion.
• It is only necessary to determine that two sets of angles are identical in order to conclude that two triangles are similar; the third set will be identical because all of the angles of a triangle always sum to 180º.
• In similar triangles, the sides of the triangles are in some proportion to one another. For example, a triangle with lengths 3, 4, and 5 has the same angle measures as a triangle with lengths 6, 8, and 10. The two triangles are similar, and all of the sides of the larger triangle are twice the size of the corresponding legs on the smaller triangle.
• If two similar triangles have sides in the ratio \(\frac{x}{y}\), then their areas are in the ratio \(\frac{x^2}{y^2}\).
OR in another way: in two similar triangles, the ratio of their areas is the square of the ratio of their sides: \(\frac{AREA}{area}=\frac{SIDE^2}{side^2}\).
For more on triangles please check Triangles chapter of the Math Book (link in my signature).
Back to original question:
The area of the right triangle ABC is 4 times greater than the area of the right triangle KLM. If the hypotenuse KL is 10 inches, what is the length of the hypotenuse AB?
(1) Angles ABC and KLM are each equal to 55 degrees --> ABC and KLM are similar triangles --> \(\frac{AREA_{ABC}}{area_{KLM}}=\frac{4}{1}\), so the sides are in ratio 2/1 --> hypotenuse KL=10 --> hypotenuse AB=2*10=20. Sufficient.
(2) LM is 6 inches --> KM=8 --> \(area_{KLM}=24\) --> \(AREA_{ABC}=96\). But just knowing the are of ABC is not enough to determine hypotenuse AB. For instance: legs of ABC can be 96 and 2 OR 48 and 4 and you'll get different values for hypotenuse. Not sufficient.
Answer: A.
Hope it helps.
General Discussion
10 May 2010, 22:47
Kudos
Bookmarks
TXTDryFly
First of all Triangles ABC & KLM are right angled at C & M resp as KL is hypot..
Area of ABC = 1/2 * AC * CB = 4 * Area of KLM = 4 * 1/2 * KM * ML
Therefore AC * CB = 4 KM * ML
1. Angles ABC and KLM = 55 deg
Therefore AC = KM as similar triangles and sides opp equal angles..
Above equation becomes CB = 4 ML or CB / ML = 4 / 1 which is further = AB / KL
as triangles are similar.. As KL is 10 , therefore AB = 40.. sufficient..
2. LM = 6 inches . Therefore KM = 8 by pythogorus theorem.. ( 10^2 - 6^2 )
But no option to find out hypot AB.. Therefore , insufficient..
Ans : A..
