Question

Equilateral Triangle MNP has perimeter 12a+18b.
Line segment QR is a midsegment.
What is QR?

Answer 1

got pic?

Answer 2

No...

Answer 3

With a perimeter = 12a + 18b, that means each side has length (12a + 18b) / 3 = 4a + 6b

A midsegment creates a 30:60:90 triangle, with side lengths which are in the ratio of x:2x:sqrt(3)*x. In other words, the side opposite the 90 degree angle created by the midsegment, is one of the sides of the equilateral triangle, which we already know has length 4a + 6b. Therefore:

==> 4a + 6b = sqrt(3)*x
==> x = (4a+6b)/sqrt(3)

However, we're looking for the length of the midsegment, which is the side opposite the 60 degree angle, which is = 2x.

Therefore, QR = 2*x = 2*(4a+6b)/sqrt(3)
==> QR = (8a + 12b)/sqrt(3)