A circle with a 4-inch radius is centered at A, and a circle with a 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?

Question

A circle with a 4-inch radius is centered at A, and a circle with a 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 1

Draw the figure. Since AP and BQ are parallel, PABQ is a trapezoid.

Draw PR to intersect BQ at R.

AP=BR=4 so RQ=5

So, the sides are
PA=4
AB=PR=13
BQ=9
QP = √(5^2+13^2) = √194

Answer 2

I agree with your answers for the length of PA, AB, PR, BQ except for QP. to solve for QP you'll do the √(13^2-5^2)=12. Since PABQ is a trapezoid, AB will be one of its opposite sides which will also act as the hypotenuse of the right-angled triangle within

Answer 3

To determine the type of quadrilateral PABQ, we need to analyze its properties. From the information given, we can observe that P and Q are the points of tangency between the segment and the small and large circles, respectively. Also, the segment is a common external tangent, meaning it touches both circles but does not intersect them.

Since P and Q are the points of tangency, drawing radii from the centers of the circles to P and Q will form right angles. Let's call the center of the small circle O₁ and the center of the large circle O₂.

Next, draw segment AB, connecting the centers of the circles. Since the circles are centered at A and B, respectively, this segment will have a length of 13 inches, as given in the problem.

Now, let's consider the quadrilateral PABQ.

1. Opposite sides PA and QB are parallel:
- This is because tangents drawn from a common point outside the circle are parallel.

2. Opposite sides PA and QB have equal lengths:
- This is because, in tangent circles, the lengths of the tangents from a common external point to each circle are equal.

3. Side AB connects the centers of the circles:
- The segment AB is the line that joins the centers of the circles, and it always passes through the point of tangency.
- This means that segment AB is a straight line and its length is given as 13 inches.

Based on these properties, we can conclude that quadrilateral PABQ is a trapezoid. Specifically, it is an isosceles trapezoid since opposite sides PA and QB are parallel and have equal lengths.

Now, let's determine the lengths of the sides of the trapezoid PABQ:

- Side PA: This is the tangent from point P to the small circle. Its length can be calculated using the Pythagorean theorem in right triangle PO₁A, where radius OA = 4 inches.
- PA^2 + OA^2 = OP^2 (by the Pythagorean theorem)
- PA^2 + 4^2 = 4^2 (since OP is the radius of the small circle)
- PA^2 = 0
- PA = 0

- Side QB: This is the tangent from point Q to the large circle. Its length can also be calculated using the Pythagorean theorem in right triangle QB¦O₂A, where radius OB = 9 inches.
- QB^2 + OB^2 = OQ^2 (by the Pythagorean theorem)
- QB^2 + 9^2 = 9^2 (since OQ is the radius of the large circle)
- QB^2 = 0
- QB = 0

Since the lengths of both PA and QB are zero, it means that both sides are degenerate and do not form part of the trapezoid. Consequently, the trapezoid PABQ degenerates into two separate circles tangent to each other at point P.