Question

Need Help With Geometry. Please Show Work. Just 11.

Answer 1

11)

when it comes to a right-triangle, having a line cutting the right-angle and perpendicular to the opposite side, like in this case, what happens is, we end up with 3 similar triangles.

a Large one, containing the other two.

a Medium one, one of the small ones.

a Small one.

so then, we can use the ratios from all these triangles, Check the picture below.

`\bf \cfrac{Large}{Small}\qquad \qquad \cfrac{x+10}{20}=\cfrac{20}{10}\implies \cfrac{x+10}{20}=2\implies x+10=40\implies \boxed{x=30} \\\\[-0.35em] \rule{34em}{0.25pt}\\\\ \cfrac{Small}{Medium}\qquad \qquad \cfrac{10}{y}=\cfrac{y}{x}\implies \cfrac{10}{y}=\cfrac{y}{30}\implies 300=y^2\implies √(300)=y`

`\bf √(3\cdot 100)=y\implies √(3\cdot 10^2)=y\implies \boxed{10√(3)=y} \\\\[-0.35em] \rule{34em}{0.25pt}\\\\ \cfrac{Medium}{Large}\qquad \qquad \cfrac{x}{z}=\cfrac{z}{x+10}\implies \cfrac{30}{z}=\cfrac{z}{30+10}\implies \cfrac{30}{z}=\cfrac{z}{40} \\\\\\ 1200=z^2\implies √(1200)=z\implies √(3\cdot 20^2)=z\implies \boxed{20√(3)=z}`