Find the measure of NM if the perimeter of the triangle is 65.

Question

Line XM is congruent to line NM. line XM=(5x+3)° Line NX=(7x-9)° It is an isosceles triangle.

Answer 1

The picture I have in mind is an isosceles triangle MNX, with sides XM=NM, which we'll call a. The angles X and N are equal so 5x+3=7x-9. So 2x=12 and x=6. X and N are therefore 33°. Dropping a perpendicular from M to the line XN creates two right-angled triangles. XN=2acos33, because the two right-angled triangles are congruent (common side, equal angles and sides of the isosceles triangle), so XN is bisected. The perimeter of the triangle is 65 so 2a+2acos33=65. Therefore a=65/2(1+cos33)=17.676 approx. Therefore NM=XM=17.676.