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Difficulty: MediumRight Triangles and the Pythagorean Theorem

In the xyxy-plane, a circle with center at the origin passes through the point (0,8)(0, 8). A line tangent to the circle at point TT passes through the point P(15,8)P(15, 8). What is the length of segment PTPT?

Answer: 15

Answer

The length of segment PTPT is 1515.
The radius of the circle is 88 because the circle is centered at (0,0)(0,0) and passes through (0,8)(0,8). The distance from the center O(0,0)O(0,0) to point P(15,8)P(15,8) is OP=152+82=17OP = \sqrt{15^2 + 8^2} = 17. The radius OTOT is perpendicular to the tangent segment PTPT at point TT, forming a right triangle △OTP\triangle OTP with hypotenuse OPOP and legs OTOT and PTPT. Using the Pythagorean theorem, PT=172−82=15PT = \sqrt{17^2 - 8^2} = 15.

Step-by-Step Solution

1
Determine the radius of the circle
Radius r=8r = 8
The circle is centered at the origin (0,0)(0, 0) and passes through (0,8)(0, 8), so the distance from the center to this point is the radius.
2
Calculate the distance from the origin O(0,0)O(0,0) to the point P(15,8)P(15, 8)
Distance OP=17OP = 17
Using the distance formula in the coordinate plane: OP=(15−0)2+(8−0)2=225+64=17OP = \sqrt{(15-0)^2 + (8-0)^2} = \sqrt{225 + 64} = 17.
3
Apply the Pythagorean theorem to the right triangle △OTP\triangle OTP
Length PT=15PT = 15
Since the tangent line PTPT is perpendicular to the radius OTOT at the point of tangency TT, △OTP\triangle OTP is a right triangle with hypotenuse OP=17OP = 17 and leg OT=8OT = 8. Thus, PT=OP2−OT2=172−82=15PT = \sqrt{OP^2 - OT^2} = \sqrt{17^2 - 8^2} = 15.

Key Concept

Tangent lines to circles and the Pythagorean theorem in the coordinate plane
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Topics

Right Triangles and the Pythagorean Theorem

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