What is the area of the quadrilateral bounded by the lines and y = 3/4

Question

What is the area of the quadrilateral bounded by the lines and  y = \frac{3}{4}*x + 6 ,  y= \frac{3}{4}*x - 6 ,  y= -\frac{3}{4}*x + 6 ,  y = -\frac{3}{4}*x - 6  ?

(A) 48
(B) 64
(C) 96
(D) 100
(E) 140

askhere · Accepted Answer

When you try to draw it, this becomes very clear.
See attached figure.

y = 3/4 x + 6 has y intercept 6 and x intercept is -8(put y =0 in the equation and calculate for x). So the line passes through (0,6) and (-8,0)
y = 3/4 x - 6 has y intercept -6 and x intercept is 8. So the line passes through (0,-6) and (8,0)
y = -3/4 x + 6 has y intercept +6 and x intercept is +8. So the line passes through (0,6) and (8,0)
y = -3/4 x - 6 has y intercept -6 and x intercept is -8. So the line passes through (0,-6) and (-8,0)

You will see that the quadrilateral is a rhombus. One might be tempted to take it as square as all sides are equal but if you look at the diagonals,
you can find that they are unequal. So it must be a rhombus.

Area of rhombus = (d1 * d2) /2 = (12 * 16)/2 = 96

Answer  C

Bunuel · Answer

MANHATTAN GMAT OFFICIAL SOLUTION: All of these line equations are of the form y = mx + b, where m is the slope and b is the y-intercept. Two of these lines have a slope of 3/4 and are thus parallel to each other. The other two lines are parallel to one another with a slope of –3/4. Two of the lines have a y-intercept of 6 while the other two lines have a y-intercept of –6. Sketch the lines: https://gmatclub.com/forum/download/file.php?id=26859 In each quadrant, we have a triangle with the dimensions 6–8–10, a multiple of the common 3–4–5 right triangle: Area = 4(1/2*bh) = 2bh = 2*6*8 = 96. Alternatively, recognize that the quadrilateral is a rhombus (four equal sides of length 10), and use the formula for the area of a rhombus: \frac{D_1*D_2}{2} , where D indicates the length of the diagonals: The correct answer is C. 2015-06-08_1520.png

Bunuel · Answer

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SHIVANSH1102 · Answer

it forms a quad with diagonals 16 and 12. 
Hence, not a rectangle or a square. Hence rhombus.
Ar: (d1*d2)/2 = 16*12/2 = 96

Tomnovhere · Answer

How do you conclude the other side is 10 do they form right angle triangle?

Posted from my mobile device

Rohit2015 · Answer

yes ..every quadrant is rt. triangle

Karpov92 · Answer

Area of a quadrilateral = ½ | x1 – x3 y1-y3 |
 x2-x4 y2- y4
 = ½ | 0-0 6+6|
 -8-8 0-0
 =1/2 { 0*0 – (12* -16)
 =1/2 * 12 * 16 = 96 (C)

Fdambro294 · Answer

Plugging All the Intercepts for Each Line:

y = 3/4x + 6

(0 , 6) and (-8 , 0)

y = 3/4x - 6

(0 , -6) and (8 , 0)

y = (-)3/4x + 6

(0 , 6) and (8 , 0)

y = (-)3/4x - 6

(0 , -6) and (-8 , 0)

4 Vertices bound the Quadrilateral:

(-8 , 0)
(0 , 6)
(8 , 0)
(0 , -6)

Method 1: You can Add up the AREAS of the 4 Right Triangles that are formed by Origin (0 ,0) as the Right Angle

4 *   = 96

Method 2:

Rule: In a Square, the 2 Diagonal are CONGRUENT and are Perpendicular, Bisectors of each Other

While the 2 Diagonals are Perpendicular to Each Other and DO Bisect Each Other:

Diagonal 1 runs from (-8 , 0) -------> to (8 , 0) = 16 Units ----- 8 on Each Side of the Origin running on the X-Axis

Diagonal 2 runs from (0 , -6) -------> to (0 , 6) = 12 Units ---- 6 on Each Side of the Origin running on the Y-Axis

Rule: in a Rhombus, the 2 Diagonals are NON-Congruent ---- but the 2 Diagonals are Perpendicular, Bisectors of Each Other

Based on these Fact, the Quadrilateral is a Rhombus

Area of a Rhombus = (Diagonal 1) * (Diagonal 2) * (1/2)

Area of this Rhombus = (16) * (12) * (1/2) = 16 * 6 =

96

-C-

Aditya997 · Answer

Hey, can anyone please let me know whether learing geometry formulas is nrcessary for gmat focus. I've seen some formulas been used in coordinate geometry questions.

Bunuel · Answer

While specific geometry knowledge is not tested on GMAT Focus, not everything involving shapes, volumes, or areas requires specialized geometry knowledge. The area of a square or rectangle, the volume of a cube or rectangular solid, and the Pythagorean theorem are not considered specific geometry knowledge by the GMAT and can still be tested. Moreover, a question can involve shapes but test another area, such as combinations or other topics. There are several  questions involving these concepts in the GMAT Prep Focus mocks .

Some aspects of coordinate geometry are also tested on GMAT Focus under the Functions tag. However, the question above is not the type you're likely to encounter on the test. For more details, check this post:  Think Geometry is not on GMAT Focus? Think Again!

Hope it helps.

What is the area of the quadrilateral bounded by the lines and y = 3/4

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What is the area of the quadrilateral bounded by the lines and y = 3/4

Prep Toolkit

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