Practice
Arcs, Angles, and Tangents Practice
Fifty original questions on circle anatomy, arcs, tangent theorems, tangent triangles, and incircle tangent lengths.
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Question 1
Explanation
A radius has one endpoint at the center and one on the circle.
- Reasoning
Classify endpoints before using a circle term.
Question 2
Explanation
A chord is a straight segment with two endpoints on the circumference.
- Reasoning
Look for two boundary endpoints.
Question 3
Explanation
A tangent has one point of contact in the circle's plane.
- Reasoning
Count intersection points.
Question 4
Explanation
A secant passes through the circle and contains a chord.
- Reasoning
Count two boundary intersections.
Question 5
Explanation
The vertex location, not the side type, makes an angle central.
- Reasoning
Find the vertex first.
Question 6
Explanation
A minor arc has the same degree measure as its central angle, so the arc is 28°.
- Reasoning
Match a central angle directly to its intercepted minor arc.
Question 7
Explanation
A minor arc has the same degree measure as its central angle, so the arc is 46°.
- Reasoning
Match a central angle directly to its intercepted minor arc.
Question 8
Explanation
A minor arc has the same degree measure as its central angle, so the arc is 73°.
- Reasoning
Match a central angle directly to its intercepted minor arc.
Question 9
Explanation
A minor arc has the same degree measure as its central angle, so the arc is 105°.
- Reasoning
Match a central angle directly to its intercepted minor arc.
Question 10
Explanation
A minor arc has the same degree measure as its central angle, so the arc is 138°.
- Reasoning
Match a central angle directly to its intercepted minor arc.
Question 11
Explanation
A minor arc has the same degree measure as its central angle, so the arc is 164°.
- Reasoning
Match a central angle directly to its intercepted minor arc.
Question 12
Explanation
Subtract the minor arc from 360°: 360-32=328.
- Reasoning
Use the full-circle total.
Question 13
Explanation
Subtract the minor arc from 360°: 360-61=299.
- Reasoning
Use the full-circle total.
Question 14
Explanation
Subtract the minor arc from 360°: 360-94=266.
- Reasoning
Use the full-circle total.
Question 15
Explanation
Subtract the minor arc from 360°: 360-127=233.
- Reasoning
Use the full-circle total.
Question 16
Explanation
Arc addition gives 41+63=104 degrees.
- Reasoning
Add adjacent arcs along the requested route.
Question 17
Explanation
Arc addition gives 58+76=134 degrees.
- Reasoning
Add adjacent arcs along the requested route.
Question 18
Explanation
Arc addition gives 92+37=129 degrees.
- Reasoning
Add adjacent arcs along the requested route.
Question 19
Explanation
Arc addition gives 115+48=163 degrees.
- Reasoning
Add adjacent arcs along the requested route.
Question 20
Explanation
Arc addition gives 126+71=197 degrees.
- Reasoning
Add adjacent arcs along the requested route.
Question 21
Explanation
Arc addition gives 33+149=182 degrees.
- Reasoning
Add adjacent arcs along the requested route.
Question 22
Explanation
Arc addition gives 84+96=180 degrees.
- Reasoning
Add adjacent arcs along the requested route.
Question 23
Explanation
Arc addition gives 57+121=178 degrees.
- Reasoning
Add adjacent arcs along the requested route.
Question 24
Explanation
Set the two tangent lengths equal; solving 2x+7=5x-20 gives x=9. The common length is 25.
- Reasoning
Tangents with one exterior endpoint are congruent.
Question 25
Explanation
Set the two tangent lengths equal; solving 4x-3=2x+13 gives x=8. The common length is 29.
- Reasoning
Tangents with one exterior endpoint are congruent.
Question 26
Explanation
Set the two tangent lengths equal; solving 6x+1=3x+22 gives x=7. The common length is 43.
- Reasoning
Tangents with one exterior endpoint are congruent.
Question 27
Explanation
Set the two tangent lengths equal; solving 7x-9=3x+19 gives x=7. The common length is 40.
- Reasoning
Tangents with one exterior endpoint are congruent.
Question 28
Explanation
Set the two tangent lengths equal; solving 5x+4=8x-20 gives x=8. The common length is 44.
- Reasoning
Tangents with one exterior endpoint are congruent.
Question 29
Explanation
Set the two tangent lengths equal; solving 9x-11=5x+17 gives x=7. The common length is 52.
- Reasoning
Tangents with one exterior endpoint are congruent.
Question 30
Explanation
Set the two tangent lengths equal; solving 3x+16=7x-12 gives x=7. The common length is 37.
- Reasoning
Tangents with one exterior endpoint are congruent.
Question 31
Explanation
Set the two tangent lengths equal; solving 11x-18=6x+17 gives x=7. The common length is 59.
- Reasoning
Tangents with one exterior endpoint are congruent.
Question 32
Explanation
Radius OT is perpendicular to tangent PT, so PT²+5²=13² and PT=12.
- Reasoning
Create the right triangle at the point of tangency.
Question 33
Explanation
Radius OT is perpendicular to tangent PT, so PT²+8²=17² and PT=15.
- Reasoning
Create the right triangle at the point of tangency.
Question 34
Explanation
Radius OT is perpendicular to tangent PT, so PT²+7²=25² and PT=24.
- Reasoning
Create the right triangle at the point of tangency.
Question 35
Explanation
Radius OT is perpendicular to tangent PT, so PT²+9²=15² and PT=12.
- Reasoning
Create the right triangle at the point of tangency.
Question 36
Explanation
Radius OT is perpendicular to tangent PT, so PT²+12²=20² and PT=16.
- Reasoning
Create the right triangle at the point of tangency.
Question 37
Explanation
Radius OT is perpendicular to tangent PT, so PT²+20²=29² and PT=21.
- Reasoning
Create the right triangle at the point of tangency.
Question 38
Explanation
Radius OT is perpendicular to tangent PT, so PT²+11²=61² and PT=60.
- Reasoning
Create the right triangle at the point of tangency.
Question 39
Explanation
Radius OT is perpendicular to tangent PT, so PT²+16²=34² and PT=30.
- Reasoning
Create the right triangle at the point of tangency.
Question 40
Explanation
Radius OT is perpendicular to tangent PT, so PT²+24²=26² and PT=10.
- Reasoning
Create the right triangle at the point of tangency.
Question 41
Explanation
Each vertex contributes its tangent length twice, so the perimeter is 2(7+9+11)=54.
- Reasoning
Pair tangent pieces by their shared exterior vertex.
Question 42
Explanation
Each vertex contributes its tangent length twice, so the perimeter is 2(6+10+13)=58.
- Reasoning
Pair tangent pieces by their shared exterior vertex.
Question 43
Explanation
Each vertex contributes its tangent length twice, so the perimeter is 2(8+12+15)=70.
- Reasoning
Pair tangent pieces by their shared exterior vertex.
Question 44
Explanation
Each vertex contributes its tangent length twice, so the perimeter is 2(5+14+16)=70.
- Reasoning
Pair tangent pieces by their shared exterior vertex.
Question 45
Explanation
Each vertex contributes its tangent length twice, so the perimeter is 2(9+13+17)=78.
- Reasoning
Pair tangent pieces by their shared exterior vertex.
Question 46
Explanation
The right triangle gives r²+24²=25², so r=7. Tangents from P are congruent, so PB=24.
- Reasoning
Combine tangent perpendicularity with congruent tangent segments.
Question 47
Explanation
The two arcs total 360: 10x+40=360, so x=32 and the minor arc is 104°.
- Reasoning
Pair corresponding minor and major arcs to make a full circle.
Question 48
Explanation
The theorem compares the two tangents drawn from one common exterior point; touching the same circle alone is insufficient.
- Reasoning
Check every theorem hypothesis, especially the shared endpoint.
Question 49
Explanation
Triangle OPT is right, so OP=sqrt(15²+8²)=17. The two tangents from P are congruent, so PU=15.
- Reasoning
Use the 8-15-17 triangle, then apply the equal-tangent theorem.
Question 50
Explanation
Their sum is 6x+60=180, so x=20. The largest arc is 3(20)+30=90 degrees.
- Reasoning
Translate semicircle into a 180-degree total before solving.
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Questions to review
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