Back to Exercise: Construct circle tangent line

Exercises: Constructing Tangent Lines From an External Point

Grade 10·21 problems·~30 min·Common Core Math - HS Geometry·standard·hsg-c-a-4

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Warm-Up

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

The radius is parallel to the tangent line.

The radius bisects the tangent line.

The radius is perpendicular to the tangent line.

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

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

45°

90°

120°

180°

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

Perpendicular bisector construction

Angle bisector construction

Parallel line construction

Equilateral triangle construction

Fluency Practice

In the tangent construction, the auxiliary circle has $OP$ as its diameter. Why must its center be at the midpoint $M$ of $OP$ ?

Because $M$ is equidistant from $T_1$ and $T_2$ .

Because a circle with diameter $OP$ has its center at the midpoint of $OP$ , which is $M$ .

Because the center must be on the original circle.

Because $M$ is the midpoint of $T_1T_2$ .

Point $T_1$ lies on the auxiliary circle (with diameter $OP$ ). By Thales' theorem, what is the measure of $\angle OT_1P$ ?

45°

60°

90°

It depends on the position of $T_1$ .

In the tangent construction for circle $O$ and external point $P$ , how many tangent lines can be drawn from $P$ to the circle?

Exactly two, one on each side of line $OP$

Exactly one, the shortest path from $P$ to the circle

Infinitely many, because $P$ is outside the circle

None, because $P$ is not on the circle

A circle has center $O$ and radius $r = 5$ . An external point $P$ is at distance $OP = 13$ from the center. Using the tangent length formula, find the length of the tangent segment $PT_1$ .

A circle has radius $r = 8$ and an external point $P$ is at distance $OP = 17$ from the center. Find the tangent length $PT_1$ .

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