RobertP wrote:In the figure bellow the triangles ABC and MNP are isosceles. AB || MN, BC || NP, the length of AC is 7 and the length of BY is 4. If the area if the white region is equal to the area of the grey region, what is the length of MP?
(A) 2√2
(B) 2√7
(C) 2√3/3
(D) 7√2/2
(E) 7√3/3
Since the sides of the two triangles are parallel and the two triangles share the same base, we know that the triangles are similar triangles.
The area of triangle ABC = bh/2 = (AC * BY)/2 = (7 * 4)/2 = 28/2 = 14.
If the area of the white region is equal to area of the grey region, then the area of each region is half of the area of triangle ABC.
Area of each region = 14/2 = 7
So the area of the white region, triangle MNP, is 7.
The proportions of the sides and heights of similar triangles are all the same.
b/h of ABC = 7/4
So b/h of MNP is also 7/4.
So area of MNP = 7 = (7x * 4x)/2
14 = 7x * 4x
14 = 28x²
1/2 = x²
x = √(1/2) = √2/2
MP = 7x = 7√2/2
The correct answer is
D.
