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In the figure, ABCD is a rectangle, and F and E are points o
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17 Mar 2019, 12:37
17 Mar 2019, 12:37
Expert Reply
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88% (01:35) correct
11% (02:45) wrong based on 43 sessions
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In the figure, ABCD is a rectangle, and F and E are points on AB and BC, respectively. The area of \(∆DFB\) is 9 and the area of ∆BED is 24. What is the perimeter of the rectangle?
(A) 18
(B) 23
(C) 30
(D) 42
(E) 48
Re: In the figure, ABCD is a rectangle, and F and E are points o
[#permalink]
20 Mar 2019, 09:41
20 Mar 2019, 09:41
1
Expert Reply
The trick to this one is to be able to identify the appropriate bases and heights of the triangles. In non-right triangles, if the base is one of the sides, then the height will not be one of the other sides.
Note that in the triangle DFB, the base, FB, is 2, and the height is AD.
The area, 9, is equal to one half of the base times the height. Hence:
\(\frac{2AD}{2} = 9\)
\(AD = 9\)
By the same logic, the base of triangle BED is 4 and the height is DC. The area is equal to 24, so:
\(\frac{4DC}{2} = 24\)
Multiply both sides by 2:
\(4DC = 48\)
And divide by 4:
\(DC = 12\)
We now have the length and width of the rectangle, and can find the perimeter, which is AD + BC + DC + AB.
However, AD = BC, and DC = AB, so:
AD + AD + DC + DC
Plug in our values:
9 + 9 + 12 + 12 = 30
Note that in the triangle DFB, the base, FB, is 2, and the height is AD.
The area, 9, is equal to one half of the base times the height. Hence:
\(\frac{2AD}{2} = 9\)
\(AD = 9\)
By the same logic, the base of triangle BED is 4 and the height is DC. The area is equal to 24, so:
\(\frac{4DC}{2} = 24\)
Multiply both sides by 2:
\(4DC = 48\)
And divide by 4:
\(DC = 12\)
We now have the length and width of the rectangle, and can find the perimeter, which is AD + BC + DC + AB.
However, AD = BC, and DC = AB, so:
AD + AD + DC + DC
Plug in our values:
9 + 9 + 12 + 12 = 30
Re: In the figure, ABCD is a rectangle, and F and E are points o
[#permalink]
14 Jun 2020, 18:42
14 Jun 2020, 18:42
it is 42, not 30
