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A triangle of area 30 is formed by the line x/c + y/(c+7) -

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A triangle of area 30 is formed by the line x/c + y/(c+7) -

by fskilnik@GMATH » Tue Mar 05, 2019 7:30 am

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GMATH practice exercise (Quant Class 20)

A triangle of area 30 is formed by the line x/c + y/(c+7) - 1 = 0 (where c is a positive constant) and the coordinate axes. What is the perimeter of this triangle?

(A) 12
(B) 24
(C) 30
(D) 36
(E) 52

Answer: [spoiler]_____(C)__[/spoiler]
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by GMATGuruNY » Tue Mar 05, 2019 8:07 am
fskilnik@GMATH wrote:GMATH practice exercise (Quant Class 20)

A triangle of area 30 is formed by the line x/c + y/(c+7) - 1 = 0 (where c is a positive constant) and the coordinate axes. What is the perimeter of this triangle?

(A) 12
(B) 24
(C) 30
(D) 36
(E) 52


Given that the area and the answer choices are all integers, the triangle is probably a Pythagorean Triple.
The only Pythagorean Triple with an area of 30 is 5-12-13.

In the given equation, test c=5, with the result that c+7 = 12:
x/5 + y/12 - 1 = 0

If x=0, then y=12, implying a y-intercept at (0, 12).
If y=0, then x=5, implying an x-intercept at (5, 0).
Implication:
In combination with the two axes, the line above will form a right triangle that has a height of 12 along the y-axis, a base of 5 along the x-axis, and a hypotenuse of 13, with the result that the area = (1/2)(12)(5) = 30.
Perimeter = 5+12+13 = 30.

The correct answer is C.
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by fskilnik@GMATH » Tue Mar 05, 2019 1:23 pm
fskilnik@GMATH wrote:GMATH practice exercise (Quant Class 20)

A triangle of area 30 is formed by the line x/c + y/(c+7) - 1 = 0 (where c is a positive constant) and the coordinate axes. What is the perimeter of this triangle?

(A) 12
(B) 24
(C) 30
(D) 36
(E) 52
Yes... 5-12-13 is an important GMAT Pythagorean Triple...

$$?\,\, = \,\,\Delta \,\,{\rm{perimeter}}$$
$${S_\Delta } = 30\,\,\,\left( * \right)$$

$${x \over c} + {y \over {c + 7}} = 1\,\,\,\, \Rightarrow \,\,\,\,\left\{ \matrix{
\,x{\rm{ - intercept}} = c \hfill \cr
\,y{\rm{ - intercept}} = c + 7 \hfill \cr} \right.\,\,\,\,\,\,\,\mathop \Rightarrow \limits^{\left( * \right)} \,\,\,\,\,\,30 = {{c\left( {c + 7} \right)} \over 2}\,\,\,$$
$$c\left( {c + 7} \right) = 60\,\,\,\left[ { = 5 \cdot 12 = \left( { - 12} \right)\left( { - 5} \right)} \right]\,\,\,\,\,\,\, \Rightarrow \,\,\,\,\,\,\,c = 5\,\,\,{\rm{or}}\,\,\,c = - 12\,\,\,\,\,\mathop \Rightarrow \limits^{{\rm{stem}}} \,\,\,\,\,c = 5$$

$$\left\{ \matrix{
\,{\rm{right}}\,\,\Delta \hfill \cr
\,{\rm{legs}}\,\,5,12 \hfill \cr} \right.\,\,\,\,\,\, \Rightarrow \,\,\,\,\,{\rm{hyp}}\,\, = 13\,\,\,\,\,\,\,\, \Rightarrow \,\,\,\,\,\,\,? = 5 + 12 + 13 = 30$$


The correct answer is (C).


We follow the notations and rationale taught in the GMATH method.

Regards,
Fabio.
Fabio Skilnik :: GMATH method creator ( Math for the GMAT)
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