​ Similar and Congruent Triangles Question

Use the pythagorean theorem to find the diagonal.

25+144=169, sqrt(169)=13, so the diagonal (OR) is 13.

The opposite sides of a rectangle are congruent. OA=5, AR=12

Let's find PO.

So, OA=PO*OR

Fill in known values.

5²=PO*13.

Make PO the variable x.

25=13x

Divide by 13 on both sides.

x=25/13

25/13 = 1.9230769230769231

Round to the nearest hundredth.

x≈1.92

13-1.92 = 11.08, so the second part (PR) is 11.08

Now, to find PA we need to do the square root of 11.08*1.92.

11.08*1.92 = 21.2736, or 21.27

sqrt(21.27) = 4.6119410230400822, or approximately 4.61.

Thanks, rarinstraw.......here is a slightly different approach.....

tan(BOR)  = 5/12   ....so

arctan(5/12) = BOR  = about 22.62°

Then, because BOA is a right angle, POA  = [90 - 22.62]°  = 67.38°

So

sin (POA)  = PA / OA

sin (POA)  = PA / 5

5 sin[67.38 ] = PA = about 4.615 units = 4.62 [ rounded]