Geometry

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In triangle $PQR,$ $M$ is the midpoint of $\overline{QR}.$ Find $PM.$
PQ = 5, PR = 8, QR = 11

Rangcr897 May 3, 2024

asinus · Answer 1 · 2024-05-03T07:06:47

Find PM.

$f_Q(x)=\sqrt{5^2-x^2}\\ f_R(x)=\sqrt{8^2-(x-11)^2}\\ 25-x_P^2=64-x_P^2+22x_P-121\\ x_P=\dfrac{121+25-64}{22}\\ x_P=3.\overline{72}\\ y_P=3.333$

$P(3.\overline{72},\ 3.333)\\ M(5.5,\ 0)\\ \overline{PM}=\sqrt{(x_M-x_P)^2+y_P^2}=\sqrt{(5.5-3.727)^2+3.333^2}\\ \color{blue}\overline{PM}=3.775$

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