In a circle with center O, segment PT is tangent to the circle at point T. The distance fr… — SAT

In a circle with center $O$ , segment $PT$ is tangent to the circle at point $T$ . The distance from point $P$ to the center of the circle is $25$ . If the radius of the circle is $7$ , what is the length of segment $PT$ ?

Answer: 24

Answer

The length of segment

PT

is 24.

Because segment

PT

is tangent to the circle at point

T

, the radius

OT

is perpendicular to

PT

. This forms a right triangle

OTP

where the right angle is at vertex

T

, the legs are

OT = 7

and

PT

, and the hypotenuse is the segment from the center to the external point

OP = 25

. By the Pythagorean theorem,

OT^2 + PT^2 = OP^2

. Substituting the known lengths yields

7^2 + PT^2 = 25^2

, which simplifies to

49 + PT^2 = 625

. Subtracting

49

from both sides gives

PT^2 = 576

. Taking the square root of both sides results in

PT = 24

Step-by-Step Solution

Identify the relationship between the radius and the tangent line at the point of tangency.

The radius

OT

is perpendicular to the tangent segment

PT

, making triangle

OTP

a right triangle with a

90^\circ

angle at vertex

T

A tangent line to a circle is always perpendicular to the radius drawn to the point of tangency.

Set up the Pythagorean theorem for the right triangle

OTP

OT^2 + PT^2 = OP^2

In any right triangle, the sum of the squares of the lengths of the legs is equal to the square of the length of the hypotenuse.

Substitute the given values

OT = 7

and

OP = 25

into the equation and solve for the length of

PT

PT = 24

Substituting values gives

7^2 + PT^2 = 25^2 \implies 49 + PT^2 = 625 \implies PT^2 = 576 \implies PT = \sqrt{576} = 24

Key Concept

A line tangent to a circle is perpendicular to the radius at the point of tangency, allowing the use of the Pythagorean theorem to find unknown lengths in the resulting right triangle.

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