Circle Properties and Theorems

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Answer Key & Explanations

Circle Properties and Theorems · Grade 10 · Worksheet 3

1. Hana is designing a circular fountain for a community plaza. The fountain has a radius of 13 meters. She wants to place two decorative lights at points A and B on the circumference such that the chord AB is exactly 24 meters long. From point A, she draws a tangent line to the circle. From point B, she draws another tangent line. These two tangent lines intersect at point C outside the circle. What is the exact length of AC, the distance from point A to the intersection point C of the tangents? Answer: 31.2 meters Solution: Let O be the center of the circle. Draw radii OA and OB. Triangle OAB is isosceles with OA = OB = 13 m (radii) and AB = 24 m.
   Full step-by-step solution

   Step 1: Let O be the center of the circle. Draw radii OA and OB. Since AC is tangent at A, OA is perpendicular to AC. Similarly, OB is perpendicular to BC. Step 2: Triangle OAB is isosceles with OA = OB = 13 m (radii) and AB = 24 m. Let M be the midpoint of AB. Then OM is perpendicular to AB. In right triangle OMA, OA = 13, AM = 12. Using the Pythagorean theorem: OM^2 = OA^2 - AM^2 = 13^2 - 12^2 = 169 - 144 = 25, so OM = 5 m. Step 3: The angle AOM = angle BOM. Let angle AOM = theta. Then sin(theta) = AM/OA = 12/13 and cos(theta) = OM/OA = 5/13. Step 4: Since AC and BC are tangents from the same external point C, they are equal in length: AC = BC. Also, OC bisects angle AOB. In triangle OAC, angle OAC = 90 degrees (tangent perpendicular to radius). Step 5: In triangle OAC, we have OA = 13, and we need AC. Note that angle AOC = theta (since OC bisects angle AOB, and angle AOB = 2*theta). Step 6: In right triangle OAC, tan(theta) = opposite/adjacent = AC/OA. Therefore AC = OA * tan(theta) = 13 * (12/5) = 156/5 = 31.2 meters. The answer is 31.2 meters.

2. A circle with center (0,0) and radius 5 intersects the line y = 3 at two points. Find the distance between these intersection points. Answer: 8 Solution: Write the equation of the circle. The circle has center (0,0) and radius 5, so its equation is: x^2 + y^2 = 25. Substitute the line equation into the circle equation.
   Full step-by-step solution

   Step 1: Write the equation of the circle. The circle has center (0,0) and radius 5, so its equation is: x^2 + y^2 = 25. Step 2: Substitute the line equation into the circle equation. The line is y = 3. Substitute y = 3 into x^2 + y^2 = 25: x^2 + (3)^2 = 25 x^2 + 9 = 25. Step 3: Solve for x. x^2 = 25 - 9 x^2 = 16 x = 4 or x = -4. So the intersection points are (4, 3) and (-4, 3). Step 4: Find the distance between these two points. Both points have the same y-coordinate (y = 3), so the distance is the difference in their x-coordinates: Distance = 4 - (-4) = 4 + 4 = 8. Therefore, the distance between the intersection points is 8.

3. Sophia is designing a circular fountain for a city plaza. The fountain has a radius of 12 meters. She wants to install a straight stone path that connects two points on the circumference, forming a chord. The perpendicular distance from the center of the fountain to this chord is 9 meters. At the midpoint of the chord, she plans to place a small statue, and from that statue, she will run a straight line to the center of the fountain. What is the length of the line from the statue to the center of the fountain in meters? Answer: 9 Solution: The perpendicular from the center of a circle to a chord bisects the chord. This means the foot of the perpendicular is exactly at the midpoint of the chord. The problem states that the perpendicular distance from the center to the chord is 9 meters.
   Full step-by-step solution

   Step 1: The perpendicular from the center of a circle to a chord bisects the chord. This means the foot of the perpendicular is exactly at the midpoint of the chord. Step 2: The statue is placed at the midpoint of the chord. Step 3: The line from the statue to the center of the fountain is the perpendicular distance from the center to the chord. Step 4: The problem states that the perpendicular distance from the center to the chord is 9 meters. Step 5: Therefore, the length of the line from the statue (at the chord's midpoint) to the center is exactly 9 meters. The answer is 9.