Question

calculator
a hockey player passes a puck to his teammate by bouncing it off a wall. the angles formed when the puck hits the wall and bounces off are congruent. some of the distances are shown.
what is the distance, x, from player 1 to the wall?
enter your answer, as a decimal, in the box. round your answer to the nearest hundredth.
x = \boxed{} ft
(image shows player 1 with vertical distance x, horizontal distance 12 ft to the wall-puck contact point; player 2 has vertical distance 8 ft, horizontal distance 9 ft to that point, with right angles at the wall and congruent angles at the contact point)

Explanation:

Step1: Identify Similar Triangles

The angles of incidence and reflection are congruent, and both triangles are right - angled. So, the two right - triangles (one with height \(x\) and base \(12\) ft, and the other with height \(8\) ft and base \(9\) ft) are similar by the AA (Angle - Angle) similarity criterion.
For similar triangles, the ratios of corresponding sides are equal. So, \(\frac{x}{12}=\frac{8}{9}\)

Step2: Solve for \(x\)

Cross - multiply to get \(9x = 12\times8\)
\(9x=96\)
Then, \(x=\frac{96}{9}\approx10.67\) (rounded to the nearest hundredth)

Answer:

\(10.67\)