Circle Properties and Theorems

Grade 10 · mathematics · 92 practice problems · read aloud

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Circle Properties and Theorems

🔵 What Are Circle Theorems?

Circle theorems are rules about angles, arcs, and lines in circles. They are powerful tools for solving geometry problems without complex measurements. You'll use them to find unknown angles and prove shapes are cyclic.

🧭 Key Theorems to Know

  • Angle at Center: The angle at the center is twice the angle at the circumference subtended by the same arc. (∠AOB = 2 × ∠ACB)
  • Angles in Same Segment: Angles in the same segment are equal.
  • Angle in a Semicircle: The angle in a semicircle is a right angle (90°).
  • Opposite Angles in Cyclic Quadrilateral: Sum to 180°.
  • Tangent-Radius Property: A tangent to a circle is perpendicular to the radius at the point of contact.

📐 Step-by-Step Problem Solving

  1. Identify: Spot all radii, chords, tangents, and known angles.
  2. Label: Mark the diagram with given information.
  3. Theorem Hunt: Which theorem applies? Look for right angles, isosceles triangles, or cyclic quadrilaterals.
  4. Calculate: Work step-by-step, stating the theorem you use.
  5. Check: Does your answer make sense? Are angles in a triangle summing to 180°?

Example 1: Angle at the Center

In circle O, chord AB subtends an angle of 40° at the circumference (∠ACB). Find the angle at the center (∠AOB).

Step 1: Identify the arc AB. Both angles subtend the same arc.

Step 2: Apply the theorem: Angle at center = 2 × Angle at circumference.

Step 3: Calculate: ∠AOB = 2 × 40° = 80°.

Example 2: Cyclic Quadrilateral

In cyclic quadrilateral ABCD, ∠A = 110° and ∠B = 75°. Find ∠C.

Step 1: Recall: Opposite angles sum to 180°. So, ∠A + ∠C = 180°.

Step 2: Substitute: 110° + ∠C = 180°.

Step 3: Solve: ∠C = 180° - 110° = 70°.

⚠️ Common Mistakes

  • Misidentifying the Subtended Arc: Ensure the angles are subtended by the exact same arc.
  • Assuming Lines are Parallel/Perpendicular: Only use given information or proven facts.
  • Forgetting Isosceles Triangles: Two radii form an isosceles triangle; base angles are equal.
  • Mixing Up Theorems: The "angle at center" theorem is often confused with "angles in same segment."

💡 Tips & Tricks

  • Color Code: Use different colors to highlight arcs and their corresponding angles.
  • Acronym: Remember "C.A.S.T." for common theorems: Center (2x), Angle (Same segment), Semicircle (90°), Tangent (90°).
  • Look for Clues: A right angle often means a tangent-radius or angle in a semicircle.
  • Draw It Big: A clear, large diagram prevents confusion.

🎯 Practice Suggestions

  • Start with simple diagrams identifying one theorem.
  • Progress to complex diagrams requiring multiple theorems.
  • Create your own flashcards with a diagram on one side and the solution steps on the other.
  • Explain a theorem to a friend—teaching is the best way to learn.
  • Practice past exam questions on circle geometry.

Practice problems

6 of the 92, worked through step by step — try them before opening the answer.

1 A circle with center (0,0) has a chord from (3,4) to (6,0). Find the length of the chord.

Hint: Use the distance formula between two points on a coordinate plane to determine the length of a line segment connecting them.

Show the answer

Answer: 5

  1. Understand the problem** We have a circle centered at (0,0) with a chord from (3,4) to (6,0). We need the length of the chord. --- **
  2. Recall the distance formula** The distance between two points (x1, y1) and (x2, y2) is: Distance = sqrt( (x2 - x1)^2 + (y2 - y1)^2 ) --- **
  3. Apply the formula** Here, (x1, y1) = (3, 4) and (x2, y2) = (6, 0). x2 - x1 = 6 - 3 = 3 y2 - y1 = 0 - 4 = -4 --- **
  4. Square the differences** (3)^2 = 9 (-4)^2 = 16 --- **
  5. Sum and take square root** Sum = 9 + 16 = 25 Square root of 25 = 5 --- **
  6. Conclusion** The chord length is 5. --- **Final Answer:** 5

Let's go step-by-step. --- **

2 A circle has equation x² + y² = 169. Find the length of the chord cut off by the line y = 5.

Hint: Consider how to find the intersection points between a circle and a horizontal line, then calculate the distance between those points.

Show the answer

Answer: 24

  1. The circle equation is x² + y² = 169, so the radius is sqrt(169) = 13.
  2. The line y = 5 intersects the circle. Substitute y = 5 into the circle equation: x² + 5² = 169
  3. Calculate: x² + 25 = 169 → x² = 169 - 25 → x² = 144
  4. Take square root: x = ±12
  5. The intersection points are (12, 5) and (-12, 5)
  6. Calculate the distance between these points: distance = 12 - (-12) = 24
  7. The chord length is 24 units.

3 A circle with center at (0,0) has a chord from (3,4) to (3,-4). Find the length of the chord.

Hint: Consider the geometric properties of chords in a circle and how their position relates to the center. For a different chord, think about its perpendicular distance from the center and how that relates to its length.

Show the answer

Answer: 8

  1. Understand the problem. We have a circle centered at (0,0) and a chord from (3,4) to (3,-4). A chord is a line segment whose endpoints lie on the circle.
  2. Identify the coordinates of the endpoints. Point A = (3,4) Point B = (3,-4)
  3. Find the length of the chord. The chord is vertical because both endpoints have the same x-coordinate (x = 3). The y-coordinates are 4 and -4.
  4. Use the distance formula for vertical line segments. Length = |y2 - y1| = |(-4) - 4| = |-8| = 8.
  5. Verify with the general distance formula. Distance = sqrt((3 - 3)^2 + (-4 - 4)^2) = sqrt(0 + (-8)^2) = sqrt(64) = 8.
  6. Conclusion. The length of the chord is 8.

4 A circle with center (0,0) has radius 5. Find the length of the chord cut off by the line y = 3.

Hint: Consider the perpendicular distance from the center to the chord and use the Pythagorean theorem to find half the chord length.

Show the answer

Answer: 8

  1. Understand the problem. We have a circle centered at (0,0) with radius 5. Its equation is: x^2 + y^2 = 25. The line y = 3 cuts the circle, forming a chord. We need the length of this chord.
  2. Find the intersection points of the line and the circle. Substitute y = 3 into the circle equation: x^2 + (3)^2 = 25 x^2 + 9 = 25 x^2 = 25 - 9 x^2 = 16 x = 4 or x = -4. So the intersection points are A = (-4, 3) and B = (4, 3).
  3. Find the length of chord AB. Points A and B have the same y-coordinate (y = 3), so the chord is horizontal. The length is the difference in their x-coordinates: Length = 4 - (-4) = 4 + 4 = 8.
  4. Final answer. The length of the chord cut off by the line y = 3 is 8.

5 A circle has center at (0,0) and radius 5. Find the length of the chord cut off by the line y = 3.

Hint: Consider how a horizontal line intersects a circle centered at the origin. The chord length can be found using the distance between intersection points.

Show the answer

Answer: 8

  1. Understand the problem. We have a circle centered at (0,0) with radius 5. Its equation is: x^2 + y^2 = 25. The line y = 3 cuts the circle, forming a chord. We need the length of this chord.
  2. Find the intersection points of the line and the circle. Substitute y = 3 into the circle's equation: x^2 + (3)^2 = 25 x^2 + 9 = 25 x^2 = 25 - 9 x^2 = 16 x = 4 or x = -4. So the intersection points are A = (-4, 3) and B = (4, 3).
  3. Find the length of the chord. The chord is the line segment from A to B. Since both points have the same y-coordinate (y = 3), the chord is horizontal. The length is the difference in their x-coordinates: Length = 4 - (-4) = 4 + 4 = 8.
  4. Final answer. The length of the chord cut off by the line y = 3 is 8.

6 A circle has center at (0,0) and radius 5. Find the length of the chord cut off by the line x = 3.

Hint: Consider the perpendicular distance from the center to the chord and use the Pythagorean theorem with the radius.

Show the answer

Answer: 8

  1. The line x = 3 is vertical and 3 units from the center (0,0).
  2. The perpendicular distance from center to chord is 3 units, and the radius is 5 units.
  3. Using the Pythagorean theorem: half the chord length = sqrt(5² - 3²) = sqrt(25 - 9) = sqrt(16) = 4.
  4. Full chord length = 2 × 4 = 8.

The answer is 8.

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