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10 Apr 2020, 03:04
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Be sure to select an answer first to save it in the Error Log before revealing the correct answer (OA)!
Difficulty:
55%
(hard)
Question Stats:
60% (02:10) correct
40%
(02:32)
wrong
based on 101
sessions
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A piece of paper in the shape of an isosceles right triangle is cut along a line parallel to the hypotenuse of the triangle, leaving a smaller triangular piece. If the area of the triangle was 25 square inches before the cut, what is the new area of the triangle?
(1) The cut is made 2 inches from the hypotenuse
(2) There was a 40 percent decrease in the length of the hypotenuse of the triangle
(1) The cut is made 2 inches from the hypotenuse
(2) There was a 40 percent decrease in the length of the hypotenuse of the triangle
10 Apr 2020, 05:35
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GKomoku
What we have are two similar triangles -- ADE, the initial one, and ABC, after the cut.
The area of two similar triangles is in the ratio of the squares of their sides...
The area of ADE = \(25....\frac{1}{2}*AE^2=25....AE=5\sqrt{2}\)
(1) The cut is made 2 inches from the hypotenuse
Take triangle COE, which is again isosceles right angled triangle..CO=OE=2, so CE=\(\sqrt{2^2+2^2}=\sqrt{8}=2\sqrt{2}\)
Thus the side AC=\(5\sqrt{2}-2\sqrt{2}=3\sqrt{2}\)
Ratio of sides = \(5\sqrt{2}:3\sqrt{2}\),
so Area will be in the ratio \((5\sqrt{2})^2:(3\sqrt{2})^2=50:18=25:9\)
New area = \(25*\frac{9}{25}=9\)
(2) There was a 40 percent decrease in the length of the hypotenuse of the triangle
So sides are in the ratio =\( x:0.6x=10:6=5:3\)...
so Area will be in the ratio \((5\sqrt{2})^2:(3\sqrt{2})^2=50:18=25:9\)
New area = \(25*\frac{9}{25}=9\)
D
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10 Apr 2020, 06:30
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GKomoku
Solution
Step 1: Analyse Question Stem
- • Triangle is an isosceles right-angle triangle
- o Base = perpendicular = x(say)
o Area = \(\frac{1}{2} *x*x\)
o \(25 = \frac{1}{2}*x^2\)
- \(x = \sqrt{50} =5\sqrt{2}\)
- o We need to find any of the side after the cut.
- After all it's goining to be an isosceles right-angled triangle only
Step 2: Analyse Statements Independently (And eliminate options) – AD/BCE
Statement 1: The cut is made 2 inches from the hypotenuse.
- • Now, simply BD is a perpendicular of AC, and it will divide the AC into two equal parts
- o BDC is a right-angled triangle
- \(BC = 5*\sqrt{2}, CD = 5\)
- o Now, \(BM = 5-2 = 3\)
- \(EM =MF = BM = 3\) [EBF is an isoceles right angled triangle]
Now, \(EF = EM + MF = 3+ 3 = 6\)
This is the hypotenuse of new triangle.
Statement 2: There was 40 percentages decrease in the length of the hypotenuse.
- • New hypotenuse = \(10 – \frac{10}{100}*40 = 10 – 4 = 6\)
- o Thus, we can find the area of new triangle.
