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Grade 10 Math — Circles and tangents

Use the two tangent properties of a circle: a tangent meets the radius at a right angle, and the two tangents from an outside point are equal.

Name: ________________________

Before you start

Three straight lines have been drawn on this circle. Find out what each one is called.

123
  1. 1. Diameter
  2. 2. Chord
  3. 3. Tangent

The tangent is the only one of the three that never gets inside the circle. And a diameter is not a separate kind of thing at all — it is simply a chord that happens to pass through the center.

  1. 1.A tangent from a point P touches a circle with center O at A. The radius OA is 8 inches and the hypotenuse OP is 17 inches. How long is the tangent PA, in inches?

  2. 2.Two tangents are drawn from a point P to a circle, touching it at A and B. How does PB compare with PA?

    • a) PB is twice PA
    • b) It depends on the radius
    • c) PB is the same length as PA
    • d) PB is half of PA

  3. 3.A tangent from a point P touches a circle with center O at A. The radius OA is 5 inches and the hypotenuse OP is 13 inches. How long is the tangent PA, in inches?

  4. 4.Match each circle and outside point to the length of the tangent from that point, in inches.

    • radius 3 inches, point 5 inches from the center
    • radius 8 inches, point 10 inches from the center
    • radius 12 inches, point 13 inches from the center
    • radius 6 inches, point 10 inches from the center
    • 6
    • 8
    • 4
    • 5

  5. 5.OAPB is a quadrilateral with right angles at A and B, and its angle at P is 70°. What is its angle at O, in degrees?

  6. 6.A tangent touches a circle with center O at the point P. What is the angle between the tangent and the radius OP?

    • a) 180°
    • b) 60°
    • c) 45°
    • d) 90°

  7. 7.How many points does a tangent share with its circle?

    • a) Infinitely many
    • b) 0
    • c) 2
    • d) 1

  8. 8.A tangent PA is 24 inches long and the hypotenuse OP of triangle OAP is 25 inches. How long is the radius OA, in inches?

Answer key — Grade 10 Math — Circles and tangents

  1. 1. 15
  2. 2. c) PB is the same length as PA
  3. 3. 12
  4. 4. radius 3 inches, point 5 inches from the center → 4; radius 8 inches, point 10 inches from the center → 6; radius 12 inches, point 13 inches from the center → 5; radius 6 inches, point 10 inches from the center → 8
  5. 5. 110
  6. 6. d) 90°
  7. 7. d) 1
  8. 8. 7