Exercises: Construct circle tangent line - Common Core Math - HS Geometry

A

Warm-Up

1.

Which statement correctly describes the relationship between a tangent line and the radius at the point of tangency?

A.

The radius is parallel to the tangent line.

B.

The radius bisects the tangent line.

C.

The radius is perpendicular to the tangent line.

D.

The radius and the tangent line form a 45° angle.

2.

Thales' theorem states that an inscribed angle that subtends a diameter of a circle measures how many degrees?

A.

45°

B.

90°

C.

120°

D.

180°

3.

To find the midpoint of a segment using compass and straightedge, which construction should you use?

A.

Perpendicular bisector construction

B.

Angle bisector construction

C.

Parallel line construction

D.

Equilateral triangle construction

B

Fluency Practice

C

Varied Practice

D

Word Problems

$Circle with radius 9, external point P at distance 41 from center, tangent segment from P to tangent point T1 with right angle marked$

1.

An architect needs to find the length of a belt that runs tangent to a circular gear. The gear has center $O$ and radius $r = 9$ cm. The belt attachment point $P$ is located 41 cm from the center $O$ .

Find the length of the tangent segment from $P$ to the point of tangency on the gear. Show your work.

$Circle representing a pillar with center O and radius 6, camera at point P distance 10, tangent lines to both tangent points T1 and T2 shown with right angles marked$

2.

A security camera at point $P$ monitors a cylindrical pillar modeled as a circle with center $O$ and radius $r = 6$ m. The camera is mounted 10 m from the center of the pillar.

Find the length of the tangent line from the camera to the edge of the pillar (the tangent length $PT_1$ ).

3.

From an external point $P$ , two tangent segments $PT_1$ and $PT_2$ are drawn to a circle with center $O$ . The tangent length is $PT_1 = 24$ and the radius of the circle is $r = 7$ .

1.

Find the distance $OP$ from the external point to the center of the circle.

2.

Without doing any additional computation, state the length of $PT_2$ and explain why.

E

Error Analysis

1.

Priya is constructing tangent lines from external point $P$ to circle $O$ .
After finding midpoint $M$ of $OP$ , drawing the auxiliary circle, and locating
the intersection point $T_1$ , she draws line $PT_1$ and declares:
"I have constructed the one tangent line from $P$ to the circle. I am done."

What error did Priya make?

A.

There are exactly two tangent lines from an external point. Priya found only one and missed $T_2$ , the second intersection of the auxiliary and original circles.

B.

The auxiliary circle should be centered at $O$ , not at the midpoint $M$ .

C.

Line $PT_1$ is a secant, not a tangent, because it passes through the interior of the circle.

D.

Priya should have used the angle bisector construction instead of the perpendicular bisector.

2.

Marcus is performing the tangent construction. He says:
"To draw the auxiliary circle, I center it at $O$ with radius $OP$
so that the auxiliary circle passes through $P$ . Then the
intersection points of this auxiliary circle with the original
circle will be the tangent points."

What is wrong with Marcus's approach?

A.

Marcus should have centered the auxiliary circle at $P$ with radius $OP$ , not at $O$ .

B.

Marcus centered the auxiliary circle at $O$ , but it should be centered at the midpoint $M$ of $OP$ with radius $MO$ , so that $OP$ is the diameter and Thales' theorem applies.

C.

Marcus should draw the auxiliary circle through $O$ and $T_1$ instead of through $P$ .

D.

Marcus's approach is correct — centering at $O$ with radius $OP$ works.

F

Challenge / Extension

1.

Prove that the two tangent segments from an external point to a circle are equal in length. That is, if $T_1$ and $T_2$ are the two points of tangency and $P$ is the external point, prove that $PT_1 = PT_2$ . Clearly label each step with the theorem or property you are using.

2.