Circles Notes and Worksheets - Lindsay Bowden - Free Printable

Let’s solve each problem one by one, step by step. We’ll use the key idea: a tangent to a circle is perpendicular to the radius at the point of tangency. Also, if two segments are drawn from the same external point and both are tangent to the circle, then those two segments are equal in length.

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Problem 1:
AB is tangent to the circle. Solve for x.
We have triangle ABC, where AB = 15, BC = 17, and AC = x (radius). Since AB is tangent and AC is radius, angle CAB is 90°. So triangle ABC is a right triangle with legs AC = x and AB = 15, hypotenuse BC = 17.

Use Pythagorean theorem:
x² + 15² = 17²
x² + 225 = 289
x² = 289 - 225 = 64
x = √64 = 8

✔ Answer: 8

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Problem 2:
Is DE a tangent segment?
We’re given FD = 17 (diameter?), FE = 24, DE = 13. Point D is on the circle, E is outside. If DE were tangent, then angle FDE would be 90° (since radius to tangent is perpendicular). But FD is diameter — so if DE were tangent at D, then triangle FDE should be right-angled at D.

Check if 13² + 17² = 24²?
169 + 289 = 458
24² = 576 → Not equal.

Wait — maybe FD is not diameter? The dot is center, so FD is radius? Wait — diagram shows F to D passing through center — so FD is diameter? Actually, looking again: the line from F to D goes through the center, so FD is diameter = 17? Then radius is 8.5? That doesn’t match other numbers.

Actually, let’s re-read: points F, D, E. FD = 17, DE = 13, FE = 24. If DE were tangent at D, then angle at D between radius and DE should be 90°. But we don’t know where the center is relative to D.

Wait — perhaps the center is midpoint of FD? If FD = 17 and it passes through center, then radius is 8.5. But then distance from center to E? Too messy.

Alternative approach: Use converse of Pythagoras. If DE is tangent at D, then triangle FDE should satisfy: FD² + DE² = FE²? Only if angle at D is 90°.

But 17² + 13² = 289 + 169 = 458 ≠ 576 = 24².

So NOT a right triangle at D → DE is NOT perpendicular to radius → NOT tangent.

✔ Answer: No

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Problem 3:
Both segments are tangent to the circle. Solve for x.
From same external point, two tangents → they are equal.

So: 10x + 3 = 12x - 3
Subtract 10x: 3 = 2x - 3
Add 3: 6 = 2x
x = 3

✔ Answer: 3

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Problem 4:
JK is tangent to the circle. Solve for x.
Angle at J is 38°, JK and JL are both tangents from point J → so triangle JKL is isosceles with JK = JL.

Therefore, angles at K and L are equal. Sum of angles in triangle = 180°.

So: angle K + angle L + 38° = 180°
2 * angle L = 142° → angle L = 71°

But wait — the question asks for x°, which is labeled at L. Is that the angle inside the triangle or the angle between tangent and chord?

Looking at diagram: x° is at point L, between tangent JL and chord KL. That’s an alternate segment theorem situation? Or just triangle angle?

Actually, since JK and JL are both tangents, and K and L are points of tangency, then the lines from center to K and L are radii, perpendicular to tangents.

But simpler: triangle JKL has JK = JL → base angles equal. Angle at J is 38°, so each base angle is (180 - 38)/2 = 71°.

And x° is marked at L — likely that base angle.

✔ Answer: 71

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Problem 5:
Both segments are tangent to the circle. Solve for missing angle x°.
The figure shows a quadrilateral formed by two tangents and two radii. The angle between the two radii is 100°. The two radii are perpendicular to the tangents, so the angles at the points of tangency are 90° each.

Sum of interior angles in quadrilateral = 360°.

So: 90° + 90° + 100° + x° = 360°
280° + x° = 360°
x° = 80°

✔ Answer: 80

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Problem 6:
Is AB a tangent segment?
Given: CA = 12 (radius), AB = 16, CB = ? Wait — CB is shown as 8? No — arrow says 8 from C to B? Wait, diagram: point C (center), A on circle, B outside. CA = 12, AB = 16, and CB = ? The label “8” is along CB? That can’t be — because if CA=12 and AB=16, and angle at A is 90° if tangent, then CB should be sqrt(12²+16²)=sqrt(144+256)=sqrt(400)=20.

But here CB is labeled as 8? That must be a mistake in reading.

Wait — looking again: the “8” is written along the line from C to B, but that would mean CB=8, while CA=12 — impossible because CB must be longer than CA if B is outside.

Perhaps the “8” is part of another measurement? Or maybe it's the length from A to some point?

Actually, rereading: the diagram shows triangle CAB, with CA=12, AB=16, and CB is the hypotenuse? But there’s a label “8” near CB — perhaps it’s indicating something else?

Wait — maybe the “8” is the length from C to the point where the line meets... no.

Alternative interpretation: Perhaps CB is composed of two parts? Or maybe the “8” is a typo?

Let me think differently. If AB is tangent at A, then angle CAB = 90°, so by Pythagoras, CB should be sqrt(CA² + AB²) = sqrt(144 + 256) = sqrt(400) = 20.

But in the diagram, if CB is labeled as 8, that contradicts. Unless the “8” is not CB.

Looking back at user’s image description: “C to A is 12, A to B is 16, and there’s a label ‘8’ along CB”. But that can’t be — unless it’s not the full length.

Wait — perhaps the “8” is the length from C to the point of intersection? I think there might be confusion.

Another possibility: Maybe the “8” is the length of the segment from C to B minus something? This is ambiguous.

But let’s assume standard setup: if AB is tangent, then triangle CAB is right-angled at A, so CB must be 20. If the diagram shows CB as 8, then it’s not possible — so AB cannot be tangent.

But that seems odd. Perhaps the “8” is misread. Let me check original problem statement: “6. Is AB a tangent segment?” with diagram showing CA=12, AB=16, and CB has a label “8” — but that must be incorrect because 12 and 16 can't form a triangle with third side 8 (triangle inequality: 12+8>16? 20>16 yes; 12+16>8 yes; but 8+16>12 yes — so triangle exists, but not right-angled).

To check if AB is tangent, we need to see if angle at A is 90°. Use converse of Pythagoras: if CA² + AB² = CB², then yes.

CA=12, AB=16, CB=? If CB=20, then yes. But if CB=8, then 12² + 16² = 144+256=400, 8²=64 ≠400 → not right-angled → not tangent.

But why is CB labeled 8? Perhaps it’s a different segment. Maybe the “8” is the length from B to the point where the line from C meets... I think there’s a misinterpretation.

Wait — in many such problems, sometimes the number is placed along the line but refers to a part. Perhaps CB is divided? For example, from C to some point is 8, and rest is unknown? But not specified.

Given the information, if we take CB = 8, then clearly not tangent. But that seems too small.

Alternative: Perhaps the “8” is the length of the tangent from B to another point? I think I need to go with logic.

Assume that for AB to be tangent, angle at A must be 90°, so CB must be 20. If the diagram intends CB to be 20, but labeled wrong, or if “8” is something else.

Looking at common problems: often in such diagrams, if CA=12, AB=16, and CB=20, then it is tangent. Here, if “8” is actually meant to be part of CB, but not specified.

Perhaps the “8” is the distance from C to the foot of perpendicular or something. I think there’s ambiguity.

But let’s calculate: if AB is tangent, then CB should be 20. If the actual CB is given as 8, then no. But 8 is less than CA=12, which is impossible if B is outside the circle.

Unless B is inside, but then AB can’t be tangent.

I think the “8” might be a red herring or mislabel. Perhaps it’s the length of another segment.

Another idea: maybe the “8” is the length from A to the point where the radius meets, but that doesn’t make sense.

Let’s look at the answer logically. In most textbooks, this setup with 12, 16, 20 is classic 3-4-5 triangle scaled by 4: 12-16-20. So likely CB is 20, and the “8” is either a mistake or refers to something else.

Perhaps the “8” is the length of the segment from C to B minus the radius or something. I think for the sake of solving, if we assume that the triangle has sides 12, 16, and if CB is 20, then yes tangent. But since the problem labels “8”, and 8≠20, then no.

But that seems harsh. Let me search for similar problems.

Upon second thought, in the diagram, the “8” might be indicating the length from C to the point where the line from B touches, but it’s confusing.

Perhaps the “8” is the length of the tangent from B to the circle, but AB is already given as 16.

I think I have to conclude based on calculation. If CA=12, AB=16, and if angle at A is 90°, then CB=20. If the actual CB is not 20, then not tangent. Since the diagram shows “8” along CB, and 8<12, it’s impossible for B to be outside, so AB cannot be tangent.

But that can’t be right because then the problem is trivial.

Another possibility: the “8” is the length of the segment from C to the intersection point with AB or something. I think I need to skip and come back.

Let’s assume that the “8” is a typo and it’s supposed to be 20. But the problem says “is AB a tangent segment?” and gives numbers, so likely we need to verify.

Use coordinates. Place C at (0,0), A at (0,12) since CA=12 vertical. If AB is horizontal, then B at (16,12), then CB = distance from (0,0) to (16,12) = sqrt(256+144)=sqrt(400)=20. So if CB is 20, then yes. But if CB is labeled 8, then no.

Since the problem includes “8”, and it’s likely that CB is intended to be 20, but written as 8 by mistake, or perhaps “8” is for another purpose.

Looking at the user’s text: “C to A is 12, A to B is 16, and there’s a label ‘8’ along CB” — but in standard problems, it’s usually 20.

Perhaps the “8” is the length from B to the point of tangency for another tangent, but not specified.

I recall that in some problems, they give the whole secant or something. Here, perhaps CB is not the straight line, but I think it is.

Let’s calculate the angle. If CA=12, AB=16, CB=8, then by law of cosines, angle at A: cos(angle CAB) = (CA² + AB² - CB²)/(2*CA*AB) = (144 + 256 - 64)/(2*12*16) = (336)/(384) = 0.875, so angle = arccos(0.875) ≈ 29 degrees, not 90, so not tangent.

So if CB=8, then not tangent.

But is CB=8 reasonable? Distance from C to B is 8, but CA=12, so B is inside the circle, so AB cannot be tangent — tangent requires B outside.

So definitely not tangent.

✔ Answer: No

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Problem 7:
Solve for x. Assume any segment that appears to be tangent is tangent.
Diagram: circle, center O, radius 9.6, tangent segment of length x, and another segment from external point to circle is 5, but wait — the 5 is along the tangent? Let’s see.

Typically, from external point P, tangent to circle at T, OT = radius = 9.6, PT = x, and there’s another line from P to O, and along that line, from P to the circle is 5? Or from O to the point is 5?

The diagram shows: from center O, a radius to the point of tangency, length 9.6. From the external point, say P, the tangent segment is x, and the line from P to O has a segment labeled 5 — probably the distance from P to the circle along the line PO is 5, meaning that if O to the circle is radius 9.6, then OP = 9.6 + 5 = 14.6? Or is 5 the distance from P to the point where PO meets the circle?

Standard setup: if from external point P, tangent PT = x, radius OT = 9.6, and OP = d, then x² + 9.6² = d².

Now, what is d? The diagram shows a segment labeled 5 along OP, from P to the circle. So if the circle has radius 9.6, and from P to the near point on circle is 5, then OP = 5 + 9.6 = 14.6.

Yes.

So: x² + (9.6)² = (14.6)²
Calculate: 9.6² = 92.16
14.6² = 213.16
So x² = 213.16 - 92.16 = 121
x = √121 = 11

✔ Answer: 11

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Problem 8:
Find the missing angle measure. Assume any segment that appears to be tangent is tangent.
Diagram: circle, center O, radius to point of tangency, angle at center is 71°, and we need to find the angle at the external point, say P.

The two radii to the points of tangency form an angle of 71° at center. The two tangents from P touch at those points. The quadrilateral formed by O, the two points of tangency, and P has two right angles (at points of tangency), and angle at O is 71°, so angle at P is 360° - 90° - 90° - 71° = 109°? But that can’t be because the angle at P should be acute usually.

Wait, no: in the quadrilateral, the angles are: at each point of tangency, 90° (between radius and tangent), at center, the angle between the two radii is 71°, and at P, the angle between the two tangents.

Sum is 360°, so angle at P = 360 - 90 - 90 - 71 = 109°.

But typically, the angle between the two tangents is supplementary to the central angle? Let me think.

Actually, the triangle formed by O and the two points of tangency is isosceles with vertex angle 71°, so base angles are (180-71)/2 = 54.5° each.

Then, at each point of tangency, the radius is perpendicular to tangent, so the angle between the tangent and the line to P is 90° - 54.5° = 35.5°.

Then in the triangle formed by P and the two points of tangency, the angles at the base are each 35.5°, so angle at P is 180 - 2*35.5 = 109°.

Same as before.

But the problem asks for "the missing angle measure", and in the diagram, it's labeled x° at the external point, so x = 109.

But let me confirm with formula: the angle between two tangents from an external point is equal to 180° minus the central angle subtended by the minor arc.

Central angle is 71°, so angle between tangents is 180° - 71° = 109°. Yes.

✔ Answer: 109

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Problem 9:
The segments are tangent to the circle. Solve for x.
Two tangents from same external point, so they are equal.

So: 2x - 17 = x - 1
Subtract x: x - 17 = -1
Add 17: x = 16

Check: left side 2*16 - 17 = 32-17=15, right side 16-1=15, equal. Good.

✔ Answer: 16

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Now, compiling all answers:

1. 8
2. No
3. 3
4. 71
5. 80
6. No
7. 11
8. 109
9. 16

Final Answer:
1. 8
2. No
3. 3
4. 71
5. 80
6. No
7. 11
8. 109
9. 16