Find: x, RM, and ST

Question

M and N are the midpoints of sides RS and RT of △RST, respectively.

Given: RM = RN = 4x + 1

ST = 9x − 5

m∠R = 60°

Find: x, RM, and ST

Answer 1

Hi Alyssa,

if you draw the triangle and label the sides you will see that the smaller triangle RMN is an isosceles triangle because two of its sides are equal. So the base angles have to be congruent.

we can find the missing angles because we have measure of angle R = 60 and we know that sum of interior angles in a triangle is 180

Measure of angle RMN= measure of angle RNM= (180-60)/2

you will see that all the three angles in triangle RMN are 60 each. So it is an equilateral triangle. So the 3rd side MN is equal to the first two sides RM and RN.

so we now have MN= 4x +1

using triangle mid segment theorem, we know that the midsegment MN ( line connecting the midpoints of two sides of a triangle is midsegment) is parallel to ST and it’s length is half of the length of ST

MN = 1/2 ST or

ST = 2 MN

plug in the values

9x -5 = 2(4x+1)

solve for x , and you can find the other lengths also.

hope this helps!