Back to Tangent lines and chord properties

Tangent Lines and Chord Properties — Practice Set

Grade 10·21 problems·~35 min·Digital SAT Math·topic·sat-geotrig-circ-tangent

Work through problems with immediate feedback

Recall / Warm-Up

A line is tangent to a circle at point $P$ . A radius is drawn to point $P$ . What angle do the tangent line and radius form?

$45\degree$

$60\degree$

$90\degree$

$180\degree$

A right triangle has a hypotenuse of $10$ and one leg of $6$ . Find the length of the other leg.

Which statement correctly describes a chord of a circle?

A chord passes through the center of the circle.

A chord is tangent to the circle.

A chord has both endpoints on the circle.

A chord always bisects the circle.

Fluency Practice

A circle has center $O$ and radius $5$ . A tangent line touches the circle at point $P$ . If the distance from $O$ to a point $Q$ on the tangent line is $13$ , find $PQ$ .

A circle has radius $7$ . A tangent line from external point $P$ touches the circle at $T$ . The distance $OP = 25$ . Find the length of the tangent segment $PT$ .

Two tangent lines are drawn from external point $P$ to a circle of radius $6$ . The distance from $P$ to the center $O$ is $10$ . Find the length of each tangent segment.

A circle has radius $10$ and a chord is $8$ units from the center (perpendicular distance). Find the chord length.

Two chords intersect inside a circle. One chord is divided into segments of $6$ and $4$ . The other chord is divided into segments of $3$ and $x$ . Find $x$ .

Varied Practice

A circle has center $(0, 0)$ and radius $5$ . A tangent line at point $(3, 4)$ meets the $x$ -axis at point $Q$ . What is the length $OQ$ ?

$5$

$\dfrac{25}{3}$

$\dfrac{20}{3}$

$\dfrac{15}{4}$

A polygon is circumscribed about a circle of radius $4$ (all sides are tangent to the circle). Two adjacent sides of the polygon share tangent segments from an external vertex $P$ . One tangent segment from $P$ measures $7$ . What is the other tangent segment from $P$ ?

A chord of length $16$ is drawn in a circle of radius $10$ . What is the perpendicular distance from the center to the chord?

Two chords intersect inside a circle. The segments of one chord are $5$ and $x$ . The segments of the other chord are $4$ and $10$ . Which equation can be used to find $x$ ?

$5 + x = 4 + 10$

$5 \cdot x = 4 + 10$

$5 \cdot x = 4 \cdot 10$

$5 + x = 4 \cdot 10$

A circle has radius $13$ . A chord is located $5$ units from the center (perpendicular distance). What is the length of the chord?

Word Problems / Application

A bicycle chain runs over two circular sprockets as a tangent line. The smaller sprocket has radius $3$ cm and the larger sprocket has radius $7$ cm. The centers of the two sprockets are $20$ cm apart.

Find the length of the tangent segment of the chain between the two sprocket contact points. (For external tangency, the length is $\sqrt{d^2 - (r_2 - r_1)^2}$ , where $d$ is the distance between centers.)

A circle has center $O$ and radius $6$ . Point $P$ is outside the circle, $10$ units from the center. Two tangent lines are drawn from $P$ to the circle, touching it at points $A$ and $B$ .

Find the length of tangent segment $PA$ .

Find the distance from $P$ to the midpoint $M$ of chord $AB$ .

A circular tunnel has a radius of $5$ meters. A horizontal beam is placed inside the tunnel at a height $3$ meters above the center of the circle.

What is the length of the beam (the chord at that height)?

Error Analysis

Alex solved the following problem:

"A tangent from external point $P$ touches a circle at $T$ . The radius is $5$ and the tangent segment $PT = 5$ . Find the distance from $P$ to the center."

Alex's work:
Distance from $P$ to center $= 5 + 5 = 10$ .

What mistake did Alex make?

Alex should have used the Pythagorean theorem since the radius and tangent segment are perpendicular at $T$ .

Alex correctly added the lengths since they share point $T$ .

Alex should have multiplied $5 \times 5 = 25$ .

The tangent segment equals the radius, so Alex's answer is correct.

Jordan solved the following problem:

"A chord of length $10$ is $6$ units from the center of a circle. Find the radius."

Jordan's work:
$r^2 = 6^2 + 10^2 = 36 + 100 = 136$ , so $r = \\sqrt{136}$ .

What error did Jordan make?

Jordan used the wrong Pythagorean relationship.

Jordan used the full chord length instead of the half-chord — the correct equation is $r^2 = 6^2 + 5^2 = 61$ , so $r = \sqrt{61}$ .

Jordan should have added $6$ and $10$ to find the radius.

Jordan's arithmetic is wrong but the method is correct.

Challenge / Extension

A circle is inscribed in a right triangle with legs $6$ and $8$ and hypotenuse $10$ . Find the radius $r$ of the inscribed circle.

Explain how the equal tangent segment theorem can be used to find the perimeter of a triangle circumscribed about a circle, if you know the segments from each vertex to the points of tangency are $a$ , $b$ , and $c$ (one segment length per vertex). Provide a formula.

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