SOLUTION: a circle witha 4 inch radius is centered at A and a circle witha 9 inch radius is centered at B, where A and B are 13 inches apart. There is a segment that is tangent to the small

Click here to see ALL problems on Triangles

Question 148795: a circle witha 4 inch radius is centered at A and a circle witha 9 inch radius is centered at B, where A and B are 13 inches apart. There is a segment that is tangent to the small circle at P and to the large circle at Q. it is a common external tangent of the two circles. what kind of quadrilateral is PABQ? what are the lengths of its sides???
Answer by Earlsdon(6294) (Show Source):
You can put this solution on YOUR website!
When you draw the diagram of the situation described in this problem, you can easily see that the resulant figure is a trapezoid with sides:
Line segment AB
Line segment AP
Line segment BQ
Line segment PQ
You can show that line segments AP and BQ are parallel because the angle formed by a tangent and the radius of a circle that intersects the point of tangency, is right angle. So that line segments AP and BQ are at right angles to line segment PQ.
You can, by inspection see that the length of line segments:
AB = 13" The sum of the radii of the two circles.
Line segment AP = 4" The radius of circle A.
Line segment BQ = 9" The radius of circle B.
Line sgment PQ requires some calculation and the use of the Pythagorean theorem to find its length.
Draw a line, parallel to line segment PQ, from A to intersect the line segment BQ at point R, so that line segment AR and PQ are parallel and congruent.
You will have formed the right triangle ARB in which line segment AB is the hypotenuse.
All you have to do is find the length of line segment AR and you will have the length of line segment PQ, the fourth side of the trapezoid.
From the right triangle ARB:

AB = 13"
BR = 5"

Subtract 25 from both sides.
Take the square root of both sides.
therefore:
The lengths of the sides of the trapezoid PABQ are:
AB = 13"
AP = 4"
BQ = 9"
PQ = 12"