Question

Jeremiah uses bamboo rods to make the frame of a tailless kite. He ties three bamboo rods together to form a right triangle PQR. He then ties another rod from P that meets RQ at a right angle. Segment PS in the figure below represents this rod and it is 12 inches long.



Which of the following could be the lengths of segments QS and SR?
Answer
QS = 6 inches, SR = 24 inches
QS = 2 inches, SR = 6 inches
QS = 44 inches, SR = 100 inches
QS = 12 inches, SR = 132 inches

Answer 1

From the question, it's impossible to determine an exact value for any of the outer sides. From the geometry, you are given three triangles (PQS, PQR, PSR). For each triangle, you can use the pythagorean theorem, but you will still have 4 unknown side lengths.
This means that you have to come up with your three pythagorean equations and test each value from the answers and see which of them correctly satisfies the equations.

Answer 2

Let QP = a, PR = b, QS = c, and SR = d so that this doesn't get confusing.

This gives you the following equations.

a^2+b^2 = (c+d)^2

12^2+c^2 = a^2

d^2+12^2 = b^2

When you solve the system you get:

288 = 2*c*d = 2*QS*SR

Answer 3

Now you plug in values until the right hand side of the equation is equal to 288. Or you could simplify by noticing that 144 = c*d

Answer 4

A)QS = 6 inches, SR = 24 inches
B)QS = 2 inches, SR = 6 inches
C)QS = 44 inches, SR = 100 inches
D)QS = 12 inches, SR = 132 inches