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Circle Properties and Theorems

Grade 10 · Mathematics · Worksheet 1

  1. In a circle with center O, radius 17 cm, chord AB is drawn. The perpendicular distance from O to AB is 8 cm. Find the length of chord AB.
    Answer: ______________
  2. A circle is defined by the equation x² + y² - 6x + 4y - 12 = 0. A line with equation y = 2x - 3 intersects the circle at two points. Find the exact length of the chord formed by this intersection. Answer: ______________
  3. A circle has center at (0,0) and radius 5. Find the length of the chord cut off by the line x = 3.
    Answer: ______________
  4. In a circle with centre O, two chords AB and CD intersect at point E inside the circle. If AE = 6, EB = 16, CE = 8, find ED. Answer: ______________
  5. A circle has center at (0,0) and radius 5. Find the length of the chord cut off by the line y = 3.
    Answer: ______________
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Answer Key & Explanations

Circle Properties and Theorems · Grade 10 · Worksheet 1

  1. In a circle with center O, radius 17 cm, chord AB is drawn. The perpendicular distance from O to AB is 8 cm. Find the length of chord AB. Answer: 30 cm Solution: Let M be the midpoint of chord AB. Then OM is perpendicular to AB, and OM = 8 cm. OA is a radius, so OA = 17 cm.
    Full step-by-step solution

    Step 1: Let M be the midpoint of chord AB. Then OM is perpendicular to AB, and OM = 8 cm. OA is a radius, so OA = 17 cm. Triangle OMA is right-angled at M. Step 2: By the Pythagorean theorem: OM^2 + AM^2 = OA^2 Step 3: Substitute: 8^2 + AM^2 = 17^2 → 64 + AM^2 = 289 → AM^2 = 289 - 64 = 225 → AM = sqrt(225) = 15 cm. Step 4: Since M is the midpoint, AB = 2 × AM = 2 × 15 = 30 cm. The answer is 30 cm.

  2. A circle is defined by the equation x² + y² - 6x + 4y - 12 = 0. A line with equation y = 2x - 3 intersects the circle at two points. Find the exact length of the chord formed by this intersection. Answer: 4√5 Solution: x² - 6x + y² + 4y = 12 (x² - 6x + 9) + (y² + 4y + 4) = 12 + 9 + 4 (x - 3)² + (y + 2)² = 25 The circle has center (3, -2) and radius 5 Find the perpendicular distance from the center to the line y = 2x - 3 Rewrite the line as 2x - y - 3 = 0 Distance d = |2(3) - (-2) - 3| / sqrt(2² + (-1)²) d = |6…
    Full step-by-step solution

    Step 1: Complete the square for the circle equation x² - 6x + y² + 4y = 12 (x² - 6x + 9) + (y² + 4y + 4) = 12 + 9 + 4 (x - 3)² + (y + 2)² = 25 Step 2: The circle has center (3, -2) and radius 5 Step 3: Find the perpendicular distance from the center to the line y = 2x - 3 Rewrite the line as 2x - y - 3 = 0 Distance d = |2(3) - (-2) - 3| / sqrt(2² + (-1)²) d = |6 + 2 - 3| / sqrt(4 + 1) d = |5| / sqrt(5) d = 5/√5 = √5 Step 4: Use the Pythagorean theorem to find half the chord length Half chord = sqrt(radius² - distance²) = sqrt(25 - 5) = sqrt(20) = 2√5 Step 5: Multiply by 2 to get the full chord length Chord length = 2 × 2√5 = 4√5 The answer is 4√5.

  3. A circle has center at (0,0) and radius 5. Find the length of the chord cut off by the line x = 3. Answer: 8 Solution: The line x = 3 is vertical and 3 units from the center (0,0). The perpendicular distance from center to chord is 3 units, and the radius is 5 units.
    Full step-by-step solution

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

  4. In a circle with centre O, two chords AB and CD intersect at point E inside the circle. If AE = 6, EB = 16, CE = 8, find ED. Answer: 12 Solution: For intersecting chords, the theorem states: AE × EB = CE × ED. Substitute the given values: 6 × 16 = 8 × ED. Calculate the left side: 96 = 8 × ED.
    Full step-by-step solution

    Step 1: For intersecting chords, the theorem states: AE × EB = CE × ED. Step 2: Substitute the given values: 6 × 16 = 8 × ED. Step 3: Calculate the left side: 96 = 8 × ED. Step 4: Solve for ED: ED = 96 ÷ 8 = 12. The answer is 12.

  5. A circle has center at (0,0) and radius 5. Find the length of the chord cut off by the line y = 3. Answer: 8 Solution: 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.
    Full step-by-step solution

    Step 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. Step 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). Step 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. Step 4: Final answer. The length of the chord cut off by the line y = 3 is 8.