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Find the value of x in the figure. Assume segments are tangent

Find the value of x in the figure. Assume segments are tangent-example-1
User John Lewin
by
8.8k points

1 Answer

5 votes

Answer:

9

Explanation:

You want the radius of a circle that has a tangent of length 12 meeting an external segment of length 6.

Pythagorean theorem

The given triangle is a right triangle with legs x and 12, and hypotenuse (x+6). The Pythagorean theorem tells you the relationship is ...

(x+6)² = x² +12²

x² +12x +36 = x² +144

12x = 108 . . . . . . . . . . . . subtract (x²+36)

x = 9 . . . . . . . . . . . . . divide by 12

The value of x in the figure is 9.

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Additional comment

You can also consider what you know about Pythagorean triples. Two that might be applicable to a side length of 12 are {3, 4, 5} and {5, 12, 13}.

If the triple used here were {5, 12, 13}, that would make x=5 and the external segment 13-5=8 instead of 6.

If the triple used here were {3, 4, 5}, it would have a scale factor of 3 to become {9, 12, 15}. Then x=9 and the external segment is 15-9 = 6. This is the solution.

Another way to solve this is using the secant/tangent relation. For this, you need to extend the line 6+x across the circle. Then the relation is ...

12² = 6(6+2x) . . . . tangent² = (segment to near)·(segment to far)

12 = 3 +x . . . . . divide by 12

x = 9

User Jacqui Gurto
by
8.3k points

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