Point P is on the line y = 2x + a. The x-coordinate of P is 4. The a

Question

https://gmatclub.com/forum/download/file.php?id=52542 Point P is on the line y = 2x + a. The x-coordinate of P is 4 . The area of the shaded quadrilateral is 24 . What is the value of a ? A. 1 B. \frac{4}{3} C. 2 D. \frac{7}{6} E. 3 Picture1.png

MathRevolution · Accepted Answer

=>

When we substitute  x  with  4 , we have the point  P(4, a+8).

The area of the trapezoid is  (\frac{1}{2})(a + a + 8)*4 = 2(2a + 8) = 24  and we have  2a * 8 = 12  or  a = 2.

Therefore, C is the answer.
Answer: C

arvind910619 · Answer

https://gmatclub.com/forum/download/file.php?id=52542 
 Point  P  is on the line  y = 2x + a.  The x-coordinate of  P  is  4 . The area of the shaded quadrilateral is  24 . What is the value of  a ?

A.  1 
 
B.  \frac{4}{3} 
 
C.  2 
 
D.  \frac{7}{6} 
 
E.  3

Hi, 
Calculate the area of rectangle and triangle and equate the sum of areas to the area to the quadrilateral.

Area of rectangle = 4*a
Area of triangle is = 1/2*4*(8+a-a)= 16
4*a + 16 = 24
4*a = 8 
a = 2

nandeta · Answer

How come the sum of parallel sides is a + a + 8? I understand a+8 but how come a?

firas92 · Answer

One of the parallel side is along the Y-axis where the side is from the origin to the  Y-intercept (x=0)  of the line y=2x+a and so the length of this side is calculated from 0 to a which is  a .

And as you know the length of the other parallel side is  a+8

So the sum of the lengths of parallel sides is  a + a+8

firas92 · Answer

y coordinate of P when x=4 is 
y=2(4)+a
y=8+a

So, P=(4,8+a)

Split the total area into

1. A rectangle with length  4  and breadth  a

2. A right triangle with base  4  and height  (8+a)-a = 8

Area of the rectangle =  4a 
Area of the triangle =  \frac{1}{2}*4*8 = 16

Total Area =  4a+16  =  24

4a=24-16 
 4a=8 
 a=2

Answer is  (C)

rounakkedia172 · Answer

We know P(4,y). Based on the figure, we can construct a line from point, lets call it Q, to get a complete rectangle, the coordinates of that point , Q, will be (0,y). 
We know, slope is 2. So, for 1 unit increase in x, we have 2 units increase in y. Therefore, y coordinate of Point P will be 2 X 4(x-coordinate) = 8.
We can find the area of newly formed rectangle. 8 units(vertical distance) x 4 units (horizontal) = 32 units^2.
We have a right-angled triangle from point Q (0,8).
We have the area of the quadrilateral = 24 units^2. Area of the triangle will be= Area of rectangle - Area of quadrilateral(given in the question) = 32 - 24 = 8 units^2.
distance from point Q and a can be calculated by:
area of triangle = bh/2.
8 = 4b/2, base =6. 
6 units down from y axis (0,8) we have (0,2).
As you have already noticed, a is y-intercept value of x will be 0 and y will be 2.

Nipunh1991 · Answer

The following sum can be solved by making two equations
Let us first plot the point P as (4,b) and the positive point on the Y axis as (0,a)

1. We have been given that the area of the trapezium (quadrilateral) is 24 = ((l1 +l2) /2) * h (4)
((a + b) / 2) * 4 = 24
a+b = 12 ......Eq.1

2. We know the slope of the line from the equation of the straight line is 2
From the slope we can get another equation
(y2-y1) / (x2-x1) = 2
((x - a) / 4 - 0 ) = 2
b - a = 8......Eq.2
 Solving both Equation we get 
2a = 4
a = 2

Answer is C

TomKSH · Answer

Let y coordinate of P be b 
Triangle part of the quadrilateral is (4*(b-a) / 2, and the rest is 4a 
so the entire part is 
  + 4a = 24 
we get b+a=12 or b=12-a

Now we have P x coordinate 4 and y coordinate 12-a = P(4,12-a) 
Insert these two figures in the given equation of y=2x+a
12-a = 4(2) + a 
a = 2

Answer C

MajorOrange · Answer

Area of a trapezium = 1/2 h(a+b), where h is the height of the trapezium, and a & b are the 2 parallel sides, of differing lengths.

From the figure, you should be able to figure that 
 1. the the Point P on the line y=2x +a,
2. the point at which it cuts the Y axis(a, which is the y intercept as mentioned in the equation)
3. the point at which the perpendicular from Point P meets the X axis(4,0)
4. and the origin 
 > form a trapezium

Fitting the relevant points into the 1/2h(a+b) formula...

here, a = a (the length of the side of the trapezium on the Y axis)
 b= y (length of the perpendicular from Point P on the X axis)

h= 4  (the length from the origin the point (4,0)

Solving this equation should give you:  a+y = 12----- eq. 1

Now, from the equation of the line(y= 2x +a), by moving the terms around and by substituting x=4 , you should get an equation of the form  a-y = -8 ------ eq. 2

Solving eq.'s 1 and 2 should given you  2a = 4 --> a = 2

Thanks!

Sonia2023 · Answer

Bunuel  
Can you please explain this question??

Bunuel · Answer

I plan to answer the question in the same way as the solutions already given. Can you let me know what specific parts you want more information about?

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Point P is on the line y = 2x + a. The x-coordinate of P is 4. The a

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