In the diagram, EF is the median of trapezoid ABCD. If AB = 5x - 9, DC = x + 3, and EF = 2x + 2, what is the value of x?

well, since EF is the median, its length is the average of AB and CD. So,

2x+2 = (5x-9 + x+3)/2
2x+2 = 3x - 3
x = 5

check: yes, 12 = (16+8)/2

x = 5 EEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEZ

the median is the average of the two bases

[(5x - 9) + (x + 3)] / 2 = 2x + 2

6x - 6 = 4x + 4

splendiculus

Why did the trapezoid go to therapy? Because it couldn't find its median! Ah, good old geometry jokes...

Anyway, back to your question. Since EF is the median of trapezoid ABCD, it means that EF is the average of the two bases AB and DC. So we can set up an equation:

EF = (AB + DC) / 2

Now let's substitute the given values into the equation:

2x + 2 = (5x - 9 + x + 3) / 2

Simplifying the right side:

2x + 2 = (6x - 6) / 2

Now, let's multiply both sides by 2 to get rid of the fraction:

4x + 4 = 6x - 6

Next, let's move the variables (x) to one side and the constants to the other side:

4x - 6x = -6 - 4

-2x = -10

Now, let's solve for x by dividing both sides by -2:

x = (-10) / (-2)

x = 5

So the value of x is 5.

To find the value of x, we can use the fact that EF is the median of the trapezoid ABCD.

The median of a trapezoid connects the midpoints of the two non-parallel sides. In this case, EF is the median of trapezoid ABCD, which means that it connects the midpoints of AB and DC.

Since EF is the median, it will have the same length as the average of AB and DC. In other words, EF = (AB + DC)/2.

Now, let's substitute the given values into this equation: EF = (5x - 9 + x + 3)/2.

Simplifying this equation, we get: 2(2x + 2) = 6x - 6.

Expanding the left side, we get: 4x + 4 = 6x - 6.

Next, let's isolate the x term by subtracting 4x from both sides: 4 = 2x - 6.

Adding 6 to both sides, we get: 10 = 2x.

Finally, dividing both sides by 2, we find that x = 5.

Therefore, the value of x is 5.