Tangents, Radii and the Alternate Segment Theorem · seed 1 · A4, ink-friendly. The answer key prints on its own page for grown-ups.

Tangents and the alternate segment

Mathematics · Geometry · ages 15-16
Name ______________________   Date ____________

  1. PA and PB are tangent segments from P to the same circle. Their total length is 30 cm. Find PA in cm.

    Answer: ______________

  2. T lies on a circle with centre O. Line m passes through T and is perpendicular to OT. What follows?

    • It passes through the centre.
    • It cuts the circle twice.
    • It is tangent at point T.
    • It contains another radius.

  3. A tangent meets a radius at the point of contact. What is that angle?

    • 45 degrees
    • 90 degrees
    • 60 degrees

  4. Tangents from outside point P touch the same circle at A and B. PA is 7 cm and chord AB is 5 cm. Find the perimeter of triangle PAB in cm.

    Answer: ______________

  5. PT is tangent at T to a circle with centre O. OP is 29 cm and PT is 20 cm. Find radius OT in cm.

    Answer: ______________

  6. PB is tangent at B to a circle with centre O. PB is 9 cm and radius OB is 12 cm. Find OP.

    • 21 cm
    • 15 cm
    • 18 cm

  7. AB is a chord and C lies on the circle. Angle ACB is 128 degrees. Tangent ray AX lies opposite C across line AB. AY is the opposite tangent ray to AX. Find angle YAB.

    • 64 degrees
    • 128 degrees
    • 104 degrees
    • 52 degrees

  8. PA and PB are tangents to one circle with centre O. Right triangles OAP and OBP share hypotenuse OP and have equal radii OA and OB. Does this prove PA = PB by right-triangle congruence?

    Circle one:   True   False

  9. AB is a chord of a circle with centre O. Tangent ray AX and circle point C lie on opposite sides of line AB. Angle ACB is 30 degrees. AB lies inside right angle XAO. Find angle OAB.

    • 60 degrees
    • 30 degrees
    • 90 degrees

  10. Tangents PA and PB touch a circle with centre O at A and B. Angle APB is 44 degrees. Find the interior angle AOB of quadrilateral OAPB in degrees.

    Answer: ______________

LightMySky · lightmysky.comW1-mt_Y-Xf0BOb1C-s1

Answer key

For grown-ups. Fold this page away before handing over the rest.

Tangents and the alternate segment W1-mt_Y-Xf0BOb1C-s1

  1. 15 · The two lengths are equal, so each is 30 / 2 = 15 cm.
  2. It is tangent at point T. · A line through T on the circle, perpendicular to contact radius OT, is tangent at T.
  3. 90 degrees · The radius meets the tangent at a right angle.
  4. 19 · The tangent segments are equal, so PB = 7 cm. The perimeter is 7 + 7 + 5 = 19 cm.
  5. 21 · OT^2 = 29^2 - 20^2 = 841 - 400 = 441, so OT = 21 cm.
  6. 15 cm · OP is the hypotenuse. OP^2 = 9^2 + 12^2 = 225, so OP = 15 cm.
  7. 52 degrees · The alternate segment theorem gives angle XAB = 128 degrees. The other tangent ray gives angle YAB = 180 - 128 = 52 degrees.
  8. True · The right triangles have the same hypotenuse and equal radius legs. They are congruent, so their corresponding tangent legs are equal.
  9. 60 degrees · The alternate segment theorem gives angle XAB = 30 degrees. It and angle OAB fill the tangent-radius right angle, so angle OAB = 90 - 30 = 60 degrees.
  10. 136 · The two contact angles are right angles. Angle AOB = 360 - 90 - 90 - 44 = 136 degrees.
Worksheet · LightMySky