Circle Properties and Theorems

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Answer Key & Explanations

Circle Properties and Theorems · Grade 10 · Worksheet 3

1. Mason draws a circle with center O and radius 17 cm. Two chords AB and CD intersect at point E inside the circle. Chord AB is divided into segments AE = 7 cm and EB = 22 cm. Chord CD is divided into segments CE = 12 cm and ED = x cm. What is the value of x? Answer: x = 77/6 or approximately 12.83 cm Solution: Recall the intersecting chords theorem. When two chords intersect inside a circle, the product of the segment lengths of one chord equals the product of the segment lengths of the other chord. In this case, for chords AB and CD intersecting at E, we have AE * EB = CE * ED.
   Full step-by-step solution

   Step 1: Recall the intersecting chords theorem. When two chords intersect inside a circle, the product of the segment lengths of one chord equals the product of the segment lengths of the other chord. In this case, for chords AB and CD intersecting at E, we have AE * EB = CE * ED. Step 2: Substitute the given values into the equation: 7 * 22 = 12 * x. Step 3: Calculate the left side: 7 * 22 = 154. So 154 = 12 * x. Step 4: Solve for x by dividing both sides by 12: x = 154 / 12. Step 5: Simplify the fraction: x = 77 / 6 (divide numerator and denominator by 2). Step 6: As a decimal, x = 77 / 6 = 12.8333... cm. Final answer: x = 77/6 cm or approximately 12.83 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. In a circle with center O, two chords AB and CD intersect at point E inside the circle. If AE = 9, EB = 16, CE = 8, find ED. Answer: 18 Solution: Substitute the given values: 9 × 16 = 8 × ED. Multiply on the left: 144 = 8 × ED. Divide both sides by 8: ED = 144 ÷ 8 = 18.
   Full step-by-step solution

   Step 1: Apply the intersecting chords theorem: AE × EB = CE × ED. Step 2: Substitute the given values: 9 × 16 = 8 × ED. Step 3: Multiply on the left: 144 = 8 × ED. Step 4: Divide both sides by 8: ED = 144 ÷ 8 = 18. The answer is 18.

4. A circle with center (0,0) has 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 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 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. Step 4: Final answer. The length of the chord cut off by the line y = 3 is 8.

5. Matiu is designing a circular mirror with a radius of 34 centimeters. He wants to hang a decorative chord across the mirror that is located 16 centimeters from the center. He then plans to attach two equal-length tangent wires from the endpoints of this chord to a single hook on the wall directly above the mirror. What is the total length of the two tangent wires combined? Answer: 60 Solution: Find half the length of the chord using the Pythagorean theorem. The radius (34 cm) is the hypotenuse, and the perpendicular distance from center to chord (16 cm) is one leg.
   Full step-by-step solution

   Step 1: Find half the length of the chord using the Pythagorean theorem. The radius (34 cm) is the hypotenuse, and the perpendicular distance from center to chord (16 cm) is one leg. Half chord = sqrt(34^2 - 16^2) = sqrt(1156 - 256) = sqrt(900) = 30 cm. Full chord length = 2 * 30 = 60 cm. Step 2: The two tangent wires from the endpoints of the chord to the external hook are equal in length. In the configuration, the hook is located directly above the mirror such that the tangents meet. The triangle formed by the chord and the two tangents is isosceles. Step 3: By a known circle geometry result, when two tangents are drawn from an external point to a circle, the tangent segments are equal. In this symmetric setup, the length of each tangent segment equals half the chord length, which is 30 cm. Step 4: The total length of the two tangent wires combined = 30 + 30 = 60 cm. The answer is 60.