Math Question!!!!!!!!!!!!!!!!!!!!!

1) P is a point on an edge of a circle, centre O.
The line PQ is the tangent to the circle at P.
R is the point on OQ such that angle ORP = 90 degrees.

a) Prove that triangle QPR and triangle OPQ are similar.
OP= 5cm. PQ= 12cm.
b) Find the exact length of OR.

10 Points for best answer!!!

Both triangles are similar because they are scalene (have 3 unequal sides and angles)
QPR 12, 4 & 10cm OPQ 5, 12 & 13cm

Length of OR is 3cm

For angle ORP to be 90 degrees the triangle needs to be a 3/4/5 triangle. As we already know that OP is 5cm (the hypoteneuse), OR must be 3cm and RP 4cm.

Sounds right but I left school 31 years ago!

a) In triangles QPR and OPQ
angle Q is common, ---------------------------- 1
angle PRQ = angle OPQ = 90 ----------------- 2
hence remaining corresponding angles
angle QPR = angle QOP -------------------------- 3
By AAA postulate triangles are similar.

b)So
PR /OP = PQ/OQ or
PR/5 = 12 /OQ ----------------------------------------… 4
OQ = sq rt{12^2 + 5^2} = 13 ------------------------- 5
From 4 and 5, PR = 60/13 ------------------------ 6
Now OR = sq rt[5^2 - {(60/13)^2}] = sq rt[{25*169 - 3600}/169] = 5/13