Triangles q r s and x y z are shown. angles q s r and x z y are right angles. angles q r s and x y z are congruent. the length of y z is 9, the length of x z is 12, and the length of hypotenuse x y is 15. given △qrs ~ △xyz, what is the value of tan(q)? three-fifths three-fourths four-fifths four-thirds

Three-fourths

Step-by-step explanation:

see the attached figure to better understand the problem

we know that

∠QSR≅∠XZY ---> given problem

∠QRS≅∠XYZ ---> given problem

so

△QRS ~ △XYZ ----> by AA Similarity theorem

Remember that, if two triangles are similar, then the ratio of its corresponding sides is proportional and its corresponding angles are congruent

That means

∠Q≅∠X

∠R≅∠Y

∠S≅∠Z

In the right triangle XYZ

Find the tangent of angle X

---> opposite side angle X divided by adjacent side angle X

substitute the given values

Simplify

Remember that

∠Q≅∠X

so

therefore

---->Three-fourths