Question

Circles P and Q are externally tangent to each other and have radii 18 meters
and 32 meters, respectively. As shown, circles P and Q are externally tangent
to a line segment at points X and Y, respectively. What is XY?

Answer 1

Answer:

  48 m

Step-by-step explanation:

You want to know the distance XY between tangent points on a segment tangent to two circles that are externally tangent to each other. The circles have radii 18 m and 32 m.

Pythagorean theorem

As shown in the attachment, a segment parallel to XY joining segments PX and QY can form a right triangle whose hypotenuse is the sum of the radii, and whose short leg is the difference of the two radii. The length of XY is the unknown leg of the right triangle, and can be found using the Pythagorean theorem.

  a² +b² = c²

  14² +b² = 50²

  b² = 50² -14² = 2304

  b = √2304 = 48

The length of XY is 48 meters.

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Additional comment

In this geometry, the length XY is twice the geometric mean of the two radii. XY = 2√(18·32) = 48.