i challenge you to answer this question! CHALLENGE YOU!!?

Question

In a trapezoid the paralell sides are in the ratio k:1 where k>1. The line segment joining the midpoints of the non-parallel siedes divides the trapizoid into two regions whoses areas are in the ratio 5:2. What is the value of k? is it?
(A) 12
(B) 5/2
(C) 13
(D) 25/4
(E) 7

HA! bet you cant do that!!
if you do.. show me how you did it =D

Answer 1

If the bases are in the ratio of k:1, then we can say they have lengths of kx and x (kx and x have ratio k:1). Let's say the trapezoid has height h.

The mid-segment of the trapezoid cuts the trapezoid into two similar trapezoids, and has length (1/2)(kx + x)

The area of a trapezoid is (1/2)(height)(b1 + b2).

The top trapezoid has height h/2, and bases kx and (1/2)(kx + x). It has an area
A1 = (1/2)(h/2)(kx + (1/2)(kx + x))
= (h/4)(kx + kx/2 + x/2)
= (h/4)(3kx/2 + x/2)
= (hx/8)(3k + 1)

The bottom trapezoid has height h/2 and bases (1/2)(kx + x) and x. It has an area
A2 = (1/2)(h/2)((1/2)(kx + x) + x)
= (h/4)(kx/2 + x/2 + x)
= (h/4)(kx/2 + 3x/2)
= (hx/8)(k + 3)

If A1:A2 = 5:2, then
[(hx/8)(3k + 1)]/[(hx/8)(k + 3)] = 5/2
(3k+1)/(k+3) = 5/2
2(3k+1) = 5(k+3)
6k + 2 = 5k + 15
k = 13
(C)

Answer 2

A = (a + b)*h/2 and assume b = ka.
Since the line joining the midpoints of the non-parallel sides divides the trapezoids height by 2, the areas of the two trapezoids will be
A_1 = (a + l)*h/4 and
A_2 = (l + b)*h/4 where l is the length of the line joining the midpoints. And you know that
5A_1 = 2A_2 so 5*(a + l)*h/4 = 2*(l + b)*h/4

Now *you* do the (very simple) algebra ☺


Doug

Answer 3

Actually, it isn't difficult. Without loss of generality, let the parallel sides be horizontal, and the left side be vertical. Now drop a perpendicular from the upper right corner to the lower line [of length k]. It will intersect the midpoint line, partitioning it into a left segment of length 1 and a right segment of length p. It will furthermore intersect the lower line, partitioning it into a left segment of length 1 and a right segment of length 2p. Let the vertical distance between the top and bottom lines be 2h. Then the area of the top trapezoid is h(1+p/2) = 2, and the bottom trapezoid is h(1+3p/2) = 5. The rectangle at the right end of the lower trapezoid has area 3, so hp = 3. Each of the triangles is half this, so they are of area 1.5. This gives us top figure: rectangle, area 0.5, triangle 1.5, total 2; bottom figure: left rectangle, 0.5, right rectangle, 3.0, triangle 1.5, and now we have the areas of everything. The length of the left rectangle is 1, so the length of the right rectangle is 6, and the base of the triangle is also 6, so the total length of k is 13.

Answer 4

2*2=5

Answer 5

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Answer 6

Cleveland

Answer 7

Um...


About 25 bucks.
Give or take tax.

Depends in which State/Province you live in I guess.

Price of gas has gone down a bit...

yup, yup.

Answer 8

Homework?

Answer 9

Yes, I believe Cleveland is the right answer.

Answer 10

of course we cant.you din say what part of the trapezoid is k
ps is this your homework