Using the given information and diagram as problem 1 below p

Using the given information and diagram as problem 1 below prove the following:

as 5, as circle with radius 5, as 1. (10 pts) Point P 13 units away from the center o of a point A is shown below. The tangents PX and PY are drawn from P to the circle. A A is drawn, chosen anywhere inside minor arc Y, and the tangent line to the circle at intersecting the other tangents at points R and Q. Prove that the perimeter of APQR is 24. of a [Note: we never officially defined it, but just in case you forgot, the perimeter triangle is the sum of the lengths of its three sides. [Another note: before you do anything here, carefully read what is given, and make sure you understand exactly what\'s going on in the diagram Another damn note: the interesting thing he is that your answer does NOT depend on the exact location of point A on minor arc XY!

Solution

Length of tangents drawn from any external point of the circle are equal.

Connect OP and OX

OP =13 cm

OX =5 cm ( The line drawn from centre of the circle to the tangent is perpendicular to the tangent)

As OPX is right angled triangle , Length of PX=12 cm and similarly PY=12 cm

Now , As Q is also an external point and QX and QA are tangents . So they would also be equal .

Lets say QX=QY=a

Similarly AR =YR , say AR=YR=b

Assume PQ = c

and , PR =d

Now for triangle PQR perimeter =PQ +QR+PR

QR=a+b

=c+a +d+b

since c+a = PX=12 cm

d+b=PY=12 cm

=12+12 =24 cm