document.write( "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??? \n" ); document.write( "

Algebra.Com's Answer #109157 by Earlsdon(6294) $\"\"$ $\"About$

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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:
\n" ); document.write( "Line segment AB
\n" ); document.write( "Line segment AP
\n" ); document.write( "Line segment BQ
\n" ); document.write( "Line segment PQ
\n" ); document.write( "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.
\n" ); document.write( "You can, by inspection see that the length of line segments:
\n" ); document.write( "AB = 13\" The sum of the radii of the two circles.
\n" ); document.write( "Line segment AP = 4\" The radius of circle A.
\n" ); document.write( "Line segment BQ = 9\" The radius of circle B.
\n" ); document.write( "Line sgment PQ requires some calculation and the use of the Pythagorean theorem to find its length.
\n" ); document.write( "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.
\n" ); document.write( "You will have formed the right triangle ARB in which line segment AB is the hypotenuse.
\n" ); document.write( "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.
\n" ); document.write( "From the right triangle ARB:
\n" ); document.write( " $\"%28AB%29%5E2+=+%28AR%29%5E2%2B%28BR%29%5E2\"$
\n" ); document.write( "AB = 13\"
\n" ); document.write( "BR = 5\"
\n" ); document.write( " $\"13%5E2+=+%28AR%29%5E2%2B5%5E2\"$
\n" ); document.write( " $\"169+=+%28AR%29%5E2%2B25\"$ Subtract 25 from both sides.
\n" ); document.write( " $\"144+=+%28AR%29%5E2\"$ Take the square root of both sides.
\n" ); document.write( " $\"AR+=+12\"$ therefore: $\"PQ+=+12\"$
\n" ); document.write( "The lengths of the sides of the trapezoid PABQ are:
\n" ); document.write( "AB = 13\"
\n" ); document.write( "AP = 4\"
\n" ); document.write( "BQ = 9\"
\n" ); document.write( "PQ = 12\"
\n" ); document.write( " \n" ); document.write( "

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