In AOPQ, the measure of Q=90°, OQ = 24, QP = 7, and PO = 25. What is the value of the tangent of ZO to the nearest hundredth?

answer is x<

47 - 90 = 43

43 - 180 = x

x = 137

we have three similar triangles, because each has a right angle and shares an angle.   let's write the angles in order: opposite to short leg, long leg, hypotenuse.

cab similar to bad similar to cbd

or as ratios,

ca: ab: cb = ba: ad: bd = cb: bd: cd

we also know

ac = ad + cd

(ad+cd): ab: cb = ba: ad: bd = cb: bd: cd

(ad+cd)/cb=cb/cd

we have cb=6, ad=5 and seek x=cd.

we reject the negative root and conclude x=4

answer: 4