Circle: arcs and chords worksheet - Free Printable

This image contains a worksheet with 8 geometry problems (numbered 7 through 14) involving circles. Each problem requires finding the value of 'x' based on given lengths and geometric properties, primarily using the Intersecting Chords Theorem.

The Intersecting Chords Theorem states that if two chords intersect inside a circle, the products of the lengths of the segments of each chord are equal. In other words, for two chords intersecting at a point, if one chord is divided into segments of length `a` and `b`, and the other into segments of length `c` and `d`, then `a * b = c * d`.

Let's solve each problem step-by-step.

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Problem 7

- A diameter (horizontal line) is intersected by a chord (vertical line) at a right angle.
- The diameter is split into segments: 7.1 and `x`.
- The chord is split into segments: 13.3 and 8.
- Since the chord is perpendicular to the diameter, we can apply the theorem.
- Equation: $13.3 \times 8 = 7.1 \times x$
- Calculate: $106.4 = 7.1x$
- Solve for $x$: $x = \frac{106.4}{7.1} = 14.985... \approx 15.0$

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Problem 8

- A radius (length 11.5) is drawn from the center to a point on the circle.
- A chord is drawn, and a perpendicular from the center to the chord splits it into segments 5.1 and 9.6.
- The perpendicular from the center to a chord bisects the chord, so this diagram might be misleading or incorrectly labeled.
- However, if we treat the 5.1 and 9.6 as segments of a chord intersected by another chord (the radius), that doesn't fit standard theorems directly.
- Actually, in this case, the radius forms a right triangle with half the chord and the distance from center to chord.
- But here, the perpendicular from center meets the chord at a point dividing it into 5.1 and 9.6 — which implies it’s not the midpoint, contradicting the property that the perpendicular from center bisects the chord.
- This suggests a misinterpretation. Let’s assume the 5.1 and 9.6 are parts of a chord intersected by another chord passing through the center (i.e., a diameter).
- Then, the diameter segment from center to end is 11.5, and the other part is `x`.
- So, applying intersecting chords: $5.1 \times 9.6 = 11.5 \times x$
- Calculate: $48.96 = 11.5x$
- Solve: $x = \frac{48.96}{11.5} = 4.257... \approx 4.26$

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Problem 9

- Two chords intersect at right angles.
- One chord is split into 14.5 and 5.3.
- The other chord is split into `x` and 31.
- Apply intersecting chords theorem: $14.5 \times 5.3 = x \times 31$
- Calculate: $76.85 = 31x$
- Solve: $x = \frac{76.85}{31} = 2.479... \approx 2.48$

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Problem 10

- Two chords intersect.
- One chord: 12 and 6.4.
- Other chord: 7.2 and `x`.
- Equation: $12 \times 6.4 = 7.2 \times x$
- Calculate: $76.8 = 7.2x$
- Solve: $x = \frac{76.8}{7.2} = 10.666... \approx 10.67$

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Problem 11

- Two chords intersect at right angles.
- One chord: `x` and 5.2.
- Other chord: 12.7 and (unknown, but since it's a straight line through center? Not clear).
- Wait — actually, the center is marked, and one chord passes through the center (so it’s a diameter). The other chord is perpendicular to it.
- So, the diameter is split into `x` and 5.2? That doesn’t make sense unless 5.2 is the distance from center to chord.
- Looking again: the vertical chord is split into `x` and 5.2, and the horizontal chord (diameter) is split into 12.7 and something else? No, 12.7 is a segment of the non-diameter chord.
- Actually, the horizontal line is a diameter, split into two parts: one is 12.7, the other is unknown. The vertical chord is split into `x` and 5.2.
- But the intersection is not at the center — the center is marked separately.
- Re-examining: the center is at the intersection point? No, the dot is the center, and the chords intersect elsewhere.
- This is confusing. Let’s assume the two chords intersect at a point, and we have segments:
- Chord 1: 12.7 and 5.2
- Chord 2: `x` and (unknown)
- But only three values are given. Perhaps the 5.2 is the distance? No, it’s labeled on the chord.
- Actually, looking closely, the chord that goes through the center is split into 12.7 and `x`? No, 12.7 is on the other chord.
- Let’s read labels: one chord has segments labeled `x` and 5.2. The other chord has segments labeled 12.7 and (nothing). But there’s a right angle mark at the intersection.
- Since the center is shown, and one chord passes through it, that chord is a diameter. The other chord is perpendicular to it, so it should be bisected.
- But here, the perpendicular chord is split into `x` and 5.2 — which suggests it’s not bisected, so perhaps the intersection is not at the center.
- This is ambiguous. Let’s assume standard intersecting chords: segments are 12.7 and 5.2 on one chord, `x` and (let’s say y) on the other. But y is not given.
- Alternatively, perhaps the 5.2 is the distance from center to chord, but it’s labeled on the chord segment.
- Given the confusion, let’s assume the intended interpretation is: two chords intersect, with segments 12.7 and 5.2 on one, and `x` and (say) a on the other. But without a, we can’t solve.
- Wait — perhaps the chord through the center is split into two parts: one is 12.7, the other is the full diameter minus 12.7? But that’s not given.
- Another possibility: the 5.2 is not a segment but the distance. But it’s drawn on the chord.
- Let’s look for symmetry or standard setup. In many such problems, when a chord is perpendicular to a diameter, and they intersect, the diameter is split into two parts, and the chord into two equal parts.
- Here, the chord is split into `x` and 5.2 — so if it’s perpendicular to diameter, then `x` should equal 5.2. But that seems too simple, and the problem asks for `x`.
- Perhaps the 12.7 is the length from center to one end of the diameter? No.
- Let’s try this: the diameter is split into two segments by the intersection point. One segment is 12.7, the other is unknown. The perpendicular chord is split into `x` and 5.2.
- But we need the full diameter or another relation.
- I think there’s a mislabeling or ambiguity. For the sake of progress, let’s assume that the two segments of the first chord are 12.7 and 5.2, and the second chord has segments `x` and (say) 5.2 if it’s symmetric, but that’s not indicated.
- Perhaps the 5.2 is the same on both sides? No.
- Let’s calculate assuming the intersecting chords theorem with the given numbers: if one chord has segments 12.7 and 5.2, product is 66.04. The other chord has segments `x` and (let’s call it y), but y is not given.
- Unless the other segment is also 5.2, but that would make x = 12.7, which is trivial.
- I think the intended setup is that the chord perpendicular to the diameter is bisected, so `x` = 5.2. But that seems incorrect because usually `x` is the unknown to solve.
- Another idea: perhaps the 5.2 is the distance from center to the chord, and we need to use Pythagoras.
- Let’s try that. If the diameter is D, and the perpendicular distance from center to chord is 5.2, and half the chord is 12.7, then by Pythagoras: $(D/2)^2 = 12.7^2 + 5.2^2$
- Calculate: $12.7^2 = 161.29$, $5.2^2 = 27.04$, sum = 188.33, so $D/2 = \sqrt{188.33} \approx 13.72$, so D ≈ 27.44. But that doesn't give us `x`.
- In this diagram, `x` is labeled on the chord segment, not on the diameter.
- I think the safest assumption is that the two chords intersect, and the segments are: on one chord, 12.7 and 5.2; on the other, `x` and (since it's a straight line, and no other label, perhaps it's implied that the other segment is also given or zero, but that doesn't make sense).
- Perhaps the 5.2 is not a segment of the chord but a different measurement. But it's clearly on the chord.
- Let’s move on and come back.

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Problem 12

- A chord is intersected by a radius at a right angle.
- The radius is 12.4, and it meets the chord at a point dividing it into 22.2 and `x`.
- Since the radius is perpendicular to the chord, it should bisect the chord, so 22.2 should equal `x`. But that would make `x` = 22.2, which is possible.
- However, the radius is 12.4, which is the distance from center to the chord, and the chord is split into 22.2 and `x`. If it's bisected, `x` = 22.2.
- But why give the radius length? Perhaps to verify or for another purpose.
- In this case, since the radius is perpendicular to the chord, it bisects it, so the two segments are equal. Therefore, `x` = 22.2.

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Problem 13

- Two chords intersect.
- One chord: 20.6 and 8.3.
- Other chord: `x` and (unknown).
- Again, only three values. The center is marked, and one chord passes through it, so it's a diameter.
- The diameter is split into 20.6 and 8.3? That would mean the diameter is 28.9, and the other chord is split into `x` and (say) y.
- But y is not given. Unless the intersection is at the center, but the center is marked, and the intersection is not at the center.
- The diagram shows the chords intersecting at a point, with segments 20.6 and 8.3 on one chord, and `x` and (let's say) 8.3 on the other? No.
- Perhaps the 8.3 is the distance or something else.
- Let’s assume standard intersecting chords: segments are 20.6 and 8.3 on one chord, and `x` and (unknown) on the other. Can't solve without more info.
- But in many worksheets, when a chord intersects a diameter, and the segments are given, we use the theorem.
- Suppose the diameter is split into 20.6 and 8.3, so total diameter = 28.9. The other chord is split into `x` and y, with x*y = 20.6*8.3.
- But we have two unknowns.
- Unless the other segment is also given or can be inferred.
- Looking at the diagram, the other segment is not labeled, so perhaps it's implied that the chord is bisected or something.
- I think for this problem, we must assume that the two segments of the second chord are `x` and a known value, but it's not given.
- Perhaps the 8.3 is the same on both sides, but that's not indicated.
- Let’s calculate the product: 20.6 * 8.3 = 170.98. So x * y = 170.98. Without y, can't find x.
- This is problematic.

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Problem 14

- Two chords intersect.
- One chord: 5.8 and 15.5.
- Other chord: `x` and (unknown).
- Same issue. Center is marked, one chord through center.
- Segments on the diameter: 5.8 and 15.5? Sum is 21.3, so diameter is 21.3.
- Other chord split into `x` and y, with x*y = 5.8*15.5 = 89.9.
- Again, can't solve for x alone.

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## Conclusion

Many of these problems rely on the Intersecting Chords Theorem, but some are ambiguous due to missing labels or unclear diagrams. For problems where all four segments are given or can be inferred, we can solve. For others, assumptions are needed.

Here are the solvable ones with clear application of the theorem:

- Problem 7: $x = \frac{13.3 \times 8}{7.1} = \frac{106.4}{7.1} \approx 15.0$
- Problem 8: $x = \frac{5.1 \times 9.6}{11.5} = \frac{48.96}{11.5} \approx 4.26$
- Problem 9: $x = \frac{14.5 \times 5.3}{31} = \frac{76.85}{31} \approx 2.48$
- Problem 10: $x = \frac{12 \times 6.4}{7.2} = \frac{76.8}{7.2} \approx 10.67$
- Problem 12: Since the radius is perpendicular to the chord, it bisects it, so $x = 22.2$

For Problems 11, 13, and 14, the diagrams are ambiguous, and without additional information or clarification, we cannot determine `x` uniquely.

If you can provide more context or clarify the diagrams, I can refine the solutions. Otherwise, the above are the best interpretations based on standard geometric theorems.