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# triangle problem

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In Triangle SPT, Q is on SP and R is on PT. We know PQ = 8, QS = 4, ST = 9, and PR = QR = $$5$$. Find all possible values of RT.

May 17, 2022

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Find all possible values of RT.

Hello Guest!

X-Y coordinate system

$$R(0,\sqrt{25-16})\\ R(0,3)\\ m_{PT}=tan\ \frac{3}{4}\\ PT(x)=m(x-x_R)+y_R=0.75(x-0)+3\\ PT(x)=0.75x+3$$

$$ST(x)=\sqrt{9^2-(x-8)^2}\\ \sqrt{9^2-(x-8)^2}=0.75x+3\\ 81-(x^2-16x+64)=0.5625x^2+4.5x+9\\ x\in \{-0.64,\ \color{blue}8\}\\ y\in \ \{\ 3.52\ ,\ \color{blue}9\}$$

invalid

$$\color{blue}T(8,9)\ R(0,3)\ P(-4,0)\ Q(4,0)\ S(8,0)$$

$$\overline{RT}=+\sqrt{(y_T-y_R)^2+(x_T-x_R)^2}=+\sqrt{(9-3)^2+(8-0^2)^2}\\ \color{blue}\overline{RT}=10$$

!

May 17, 2022
edited by asinus  May 17, 2022