Inscribed circle

XYZ is a right triangle with a right angle at the vertex X and an inscribed circle with a radius of 5 cm. Find the area of the triangle XYZ if XZ = 14 cm.

Correct answer:

S =  157.5 cm²

Step-by-step explanation:

$$\begin{gathered} r=5 \\ y=14 \\ s=(x+y+z)/2 \\ {x}^2=y^2+z^2 \\ S=sr \\ S=zy/2 \\ zy/2=(z+y+\sqrt{z^2+y^2})/(2r) \\ z=2\cdot (y\cdot r-r^2)/(y-2\cdot r)=2\cdot (14\cdot 5-5^2)/(14-2\cdot 5)=\frac{45}{2}=22.5 \\ x=\sqrt{y^2+z^2}=\sqrt{14^2+22.5^2}=\frac{53}{2}=26.5 \\ S=z\cdot y/2=22.5\cdot 14/2=157.5{\text{cm}}^2 \end{gathered}$$

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