Question

Triangle XYZ is an isosceles Triangle with legs XY and ZY. Segments QR and QS are perpendicular bisectors of triangle XYZ. Find the length of segment QR if QS = 1/2x + 2 and QS = x - 10.

Answer 1

we know XYZ is an isosceles, thus XY = YZ, the perpendicular segment bisectors of QR and QS are also equal to each other in length, because they both are segment bisectors and thus YR=RX=YS=SZ, so any perpendicular line stemming from the same length on each side will meet its counterpart right on the middle of the triangle.

$\bf \stackrel{QR}{\cfrac{x}{2}+2}~~=~~\stackrel{QS}{x - 10}\implies \stackrel{\textit{multiplying both sides by }\stackrel{LCD}{2}}{2\left( \cfrac{x}{2}+2 \right)=2(x-10)}\implies x+4=2x-20 \\\\\\ 4=x-20\implies 24=x \\\\[-0.35em] ~\dotfill\\\\ \stackrel{QR}{\cfrac{24}{2}+2}\implies 12+2\implies 14$