Level 2 · Intermediate ← Back to lesson

Constructing Tangents from an External Point: Practice

12 multiple-choice questions, progressively harder.

Level 2 · Intermediate 0 / 12 answered

Question 1 of 12

1

On a coordinate grid, the given circle is centered at $O=(1,7)$ . The external point is $P=(9,13)$ . Where is the center of the auxiliary circle?

Answer choices for question 1 A $(4,3)$ B $(10,20)$ C $(8,6)$ D $(5,10)$
2

The near point $A$ of the given solid circle is inside the dashed circle. The opposite point $B$ is outside. Which argument shows there are at least two crossings?

Answer choices for question 2 A Each of the two arcs of the solid circle from $A$ to $B$ must cross the dashed circle B The dashed circle touches the solid circle at $A$ and $B$ C Line $OP$ meets both circles, so the circles cross on that line D The centers lie on $OP$ , so one of them is a crossing
3

A tangent from $P$ to the given circle touches at $T$ . The original radius is $20$ cm and $PT$ is $21$ cm. What compass opening draws the auxiliary circle?

Answer choices for question 3 A $10$ cm B $14.5$ cm C $20.5$ cm D $29$ cm
4

The given circle has positive radius. Point $P$ is $4$ mm closer to $O$ than a point on the circle is. What happens to any line through $P$ ?

Answer choices for question 4 A It touches the circle once B It lies outside the circle C It crosses the circle at two distinct points D Its intersection count depends on its direction
5

The midpoint $M$ is correctly marked, but the compass is opened to less than $MO$ . Which property of the intended auxiliary circle fails?

Answer choices for question 5 A Its center is the midpoint of $OP$ B Both $O$ and $P$ are on its boundary C $O$ and $P$ are equally far from its center D The center lies on segment $OP$
6

Point $X$ is on the auxiliary circle with diameter $OP$ and differs from both endpoints. Its distance from $O$ is less than the given circle's radius. What can you conclude about line $PX$ at $X$ ?

Answer choices for question 6 A It is tangent to the given circle at $X$ B It is parallel to $OX$ C It touches the auxiliary circle at $X$ D It is perpendicular to $OX$ , and $X$ is inside the given circle
7

The original circle has radius $24$ mm. The outside point $P$ gives an auxiliary radius of $16$ mm. The ray from $O$ through $P$ crosses the original circle at $A$ . What is the distance $AP$ ?

Answer choices for question 7 A $8$ mm B $16$ mm C $32$ mm D $56$ mm
8

Point $X$ lies on a circle of radius $9$ cm centered at $O$ . The lengths of $PX$ and $OP$ are $13$ cm and $16$ cm. Is line $PX$ tangent at $X$ ?

Answer choices for question 8 A Yes, because $OP$ is longer than $OX$ B Yes, because $X$ lies on the given circle C No, because the side lengths do not make a right angle at $X$ D No, because a tangent segment must be shorter than the radius
9

The diagram shows a tangent from $P$ to the given circle at $T_1$ . The second tangent has yet to be drawn. What stays fixed when you reflect the drawing in $OP$ to find it?

Answer choices for question 9 A $T_1$ only B $O$ and $T_1$ C Both $O$ and $P$ D $P$ and $T_1$
10

The arc intersections $U$ and $V$ are visible, but the midpoint mark has been erased. Which construction recovers it directly?

Answer choices for question 10 A Draw a line from $U$ to $O$ and extend it B Draw line $UV$ and mark where it meets $OP$ C Draw a circle with center $V$ passing through $O$ D Draw a line through $V$ parallel to $OP$
11

Suppose two distinct circles share three different points. Why would this contradict the uniqueness used in the tangent construction?

Answer choices for question 11 A Every radius would need to be parallel to $OP$ B Three points would define two different triangles C The segment joining the centers would have to be tangent to both circles D The three points form a triangle that would have two circumcenters
12

The distance $OP$ is $31$ cm. Point $T_1$ has already been constructed as a contact point. Why does moving $P$ to another point that is also $31$ cm from $O$ leave the auxiliary radius unchanged?

Answer choices for question 12 A The auxiliary radius depends only on the length of the center-to-point segment B The new external point is also the old midpoint C The auxiliary center remains in the same place D The old contact points stay tangent to the new lines

← Back to lesson