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Constructing Tangents from an External Point: Practice

12 multiple-choice questions, progressively harder.

Level 3 · Challenge 0 / 12 answered

Question 1 of 12

1

A line through $P$ is tangent at $T$ . A proof needs to show that $T$ is on the circle with diameter $OP$ . Which sequence proves the claim?

Answer choices for question 1 A Use tangency to get a right angle at $P$ , then use the semicircle theorem B Use the semicircle theorem to assume that $T$ is already on the auxiliary circle C Use tangency to get the right angle at $T$ , then use the semicircle converse D Use the equal tangent lengths to show that $P$ is the center
2

The given circle has center $O=(0,0)$ and radius $5$ . The outside point is $P=(10,5)$ . Which pair can be the two contact points?

Here are the four candidates.
$\begin{gathered} A=(0,5)\\ B=(4,-3)\\ C=(3,4)\\ D=(-5,0) \end{gathered}$

Answer choices for question 2 A $A$ and $C$ B $A$ and $B$ C $B$ and $D$ D $C$ and $D$
3

A circle centered at $O$ has radius $7$ units, and $OP$ is $23$ units. An extra circle is drawn with center $O$ and radius $OP$ . How does this setup differ from the intended construction?

Answer choices for question 3 A The circles still give the two intended contact points B The two circles meet once at $P$ C They are identical and share every point D The circles are concentric with different radii, so they have no crossings
4

Point $P$ lies on the given circle. A diameter circle on $OP$ has been drawn anyway, as shown. Why can you not draw the tangent by joining $P$ to the common point?

Answer choices for question 4 A The only common point is $P$ itself, and one point cannot determine a line B The only common point is $O$ , which determines line $OP$ C There are two crossings, so the line through them is a tangent D The dashed circle has the same center as the solid circle
5

The compass opening for finding the midpoint is exactly half of $OP$ . The arcs meet once. Why does this fail as the perpendicular-bisector construction shown in the lesson?

Answer choices for question 5 A The meeting point is nearer to $O$ than to $P$ B The meeting point is off the line $OP$ C The single meeting point gives no second point to determine the bisector line D The arcs use unequal compass openings
6

One contact point is above $OP$ and the other below it. Which argument shows that the two tangent segments from $P$ have equal lengths?

Answer choices for question 6 A Both are radii of the auxiliary circle B Both triangles have their right angle at $P$ C The segments are parts of the same infinite line D Both are legs in right triangles with the same hypotenuse $OP$ and equal contact radii
7

A circle centered at $O$ has radius $10$ cm and $OP$ is $25$ cm. In a correct construction, what is the exact length of either tangent segment?

Answer choices for question 7 A $15$ cm B $5\sqrt{21}$ cm C $5\sqrt{29}$ cm D $10\sqrt{5}$ cm
8

For an outside point $P$ , the compass is centered at the correct midpoint $M$ but opened to $OP$ . If this larger circle happens to cross the given circle, can either crossing be a contact point?

Answer choices for question 8 A No. Every contact must be exactly $MO$ from $M$ , but these crossings are farther away B Yes, if either crossing lies on the original circle C Yes, both circles have the correct centers D Yes, because the larger opening still makes OP a diameter
9

Point $P$ is strictly inside the original circle and differs from $O$ . Why does the circle with diameter $OP$ have no common point with the original circle?

Answer choices for question 9 A Every point of the auxiliary circle is more than one original radius from $O$ B The two circles are concentric C The auxiliary circle has zero radius D Every point of the auxiliary circle is at most $OP$ from $O$ , and $OP$ is less than the original radius
10

The two tangents from $P$ are drawn. A point $Q$ is chosen on one tangent, with $Q$ different from both $P$ and its contact point. Can both of those same tangent lines also pass through $Q$ ?

Answer choices for question 10 A Yes, because all outside points share the same two tangents B Yes, because both lines are tangent to the same circle C No. The two distinct lines already meet at $P$ , so only one can pass through $Q$ D No, because a tangent line cannot contain another outside point
11

The circles intersect at $T_1$ and $T_2$ . A solution argues that both lines through $P$ are tangent because $T_1$ and $T_2$ lie on the given circle. What fact is missing?

Answer choices for question 11 A The angle at each contact between its radius and its line through $P$ is right B Both intersections must be a distance $OP$ from $P$ C Both intersections must lie on segment $OP$ D The tangent lines must be perpendicular to each other
12

A proposed tangent through an outside point $P$ is claimed to have the direction of line $OP$ . What actually happens to a line through $P$ in that direction?

Answer choices for question 12 A It is parallel to $OP$ and meets the circle once B It is the line $OP$ itself, so it crosses the given circle at both ends of a diameter C It is perpendicular to $OP$ at $P$ D It stays outside the circle because $P$ is outside

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