Constructing Tangents from an External Point: Practice Level 1 | Geometry

Level 1 · Foundational 0 / 12 answered

Question 1 of 12

1

The construction sketch shows the solid circle centered at $O$ and a dashed circle. Which segment is a diameter of the dashed circle?

Answer choices for question 1 A $OM$ B $OP$ C $MP$ D $OT_1$
2

The center and external point are $46$ mm apart. Which compass opening can draw midpoint arcs that cross in two places?

Answer choices for question 2 A $19$ mm B $23$ mm C $22$ mm D $25$ mm
3

In the diagram, the compass opening stayed fixed while drawing arcs from $O$ and $P$ . Why does line $UV$ bisect $OP$ ?

Answer choices for question 3 A Each arc crossing is equally far from $O$ and $P$ B $U$ and $V$ lie on the given circle C Each arc crossing lies on segment $OP$ D $UV$ has the same length as $OP$
4

The auxiliary circle already crosses the given circle at $T_1$ and $T_2$ . What is the last straightedge action?

Answer choices for question 4 A Draw line $T_1T_2$ only B Draw a line through $M$ and $T_1$ C Draw the two lines through $P$ , one to each crossing D Draw the perpendicular bisector of $T_1T_2$
5

The midpoint $M$ of $OP$ is $9$ units from $P$ . What is the diameter of the auxiliary circle used to construct the tangents?

Answer choices for question 5 A $4.5$ units B $9$ units C $18$ units D $27$ units
6

The circles in the diagram cross at $T_1$ and $T_2$ . How would the line through those two points meet the given solid circle?

Answer choices for question 6 A It would be a secant B It would touch at $T_1$ only C It would touch at $T_2$ only D It would miss the solid circle
7

Point $T$ is an intersection of the given circle and the circle with diameter $OP$ . Which angle does that diameter make right at $T$ ?

Answer choices for question 7 A $\angle TOP$ B $\angle OPT$ C $\angle OMT$ D $\angle OTP$
8

Point $P$ coincides with the center $O$ of a positive-radius circle. How many tangent lines pass through $P$ ?

Answer choices for question 8 A One B Zero C Two D Infinitely many
9

Point $P$ lies on a ray starting at $O$ , beyond that ray's crossing with the given circle. How many tangent lines through $P$ can the diameter construction produce?

Answer choices for question 9 A No tangent lines B One tangent line C One tangent line and one secant D Two distinct tangent lines
10

At the crossing $T_2$ , which segment is a radius of the given circle centered at $O$ ?

Answer choices for question 10 A $MT_2$ B $PT_2$ C $OT_2$ D $OP$
11

Point $P$ lies on the given circle. Four proposed lines through $P$ make angles of $30^\circ$ , $45^\circ$ , $60^\circ$ , and $90^\circ$ with ray $PO$ on one side of that ray. Which angle belongs to the tangent?

Answer choices for question 11 A $30^\circ$ B $90^\circ$ C $45^\circ$ D $60^\circ$
12

After both contact points are located, which tool use extends a tangent segment to a tangent line?

Answer choices for question 12 A Align an unmarked straightedge with the contact point and $P$ , then extend in both directions B Swing a compass arc from the contact point C Join the two contact points and stop at them D Trace the given circle from a contact point toward $P$