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What is the length of MP?

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What is the length of MP?

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Zoom here, https://postimg.cc/image/q41qng77v/

In the figure above, triangle ABC and MNP are both isosceles. AB is parallel to MN, BC is parallel to NP, the lenght of AC is 7 and the lenght of BY is 4. If the area of the unshaded region is equal to the area of the shaded region, what is the lenght of MP?

$$A.\ 2\sqrt{2}$$
$$B.\ 2\sqrt{7}$$
$$C.\ \frac{2\sqrt{3}}{3}$$
$$D.\ \frac{7\sqrt{2}}{2}$$
$$E.\ \frac{7\sqrt{3}}{3}$$

The OA is D.

If the area of unshaded region is equal to the area of shaded region, then the area of the big triangle is twice the area of the little triangle, right?

Then, I can say that
$$Area_{\triangle ABC}=2\cdot Area_{\triangle MNP}$$
I stuck here. I think that it should be solve it using similar triangles theory, but I don't understand how can I apply it. Experts, any suggestion, please? Thanks.

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LUANDATO wrote:


Zoom here, https://postimg.cc/image/q41qng77v/

In the figure above, triangle ABC and MNP are both isosceles. AB is parallel to MN, BC is parallel to NP, the lenght of AC is 7 and the lenght of BY is 4. If the area of the unshaded region is equal to the area of the shaded region, what is the lenght of MP?

$$A.\ 2\sqrt{2}$$
$$B.\ 2\sqrt{7}$$
$$C.\ \frac{2\sqrt{3}}{3}$$
$$D.\ \frac{7\sqrt{2}}{2}$$
$$E.\ \frac{7\sqrt{3}}{3}$$

The OA is D.

If the area of unshaded region is equal to the area of shaded region, then the area of the big triangle is twice the area of the little triangle, right?

Then, I can say that
$$Area_{\triangle ABC}=2\cdot Area_{\triangle MNP}$$
I stuck here. I think that it should be solve it using similar triangles theory, but I don't understand how can I apply it. Experts, any suggestion, please? Thanks.
You're exactly right. First, we can find the area of MNP. If triangle ABC has an area of 7*4/2 = 14, then MNP will be exactly half that, or 7.

Now, because we know that ABC and MNP are similar triangles, the ratio of their respective sides will be the same. If the Height/Base of ABC is 4/7, then the Height/Base of MNP is also 4/7.

So let's say the height of MNP is 4x, and the base of MNP is 7x, to give us that 4/7 ratio. We also know that the area of MNP is 7.

Now we can plug the above info into the area equation of a triangle: Base * Height / 2 = Area --> (7x * 4x)/2 = 7.

28x^2 = 14
x^2 = 1/2
x = 1/√ 2 = √ 2/ 2

We want the base, or MP, which is 7x. If x = √ 2/ 2, then 7x = 7 * √ 2/ 2,. The answer is D

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