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Practice: tangents and the inscribed circle

Geometry, grade 11 (4 units) Lesson 18 of 20Practice

Practice: tangents and the inscribed circle

1

PA\displaystyle PA is a tangent to the circle with centre O\displaystyle O, at A\displaystyle A.
OA=8\displaystyle {OA}={8} and OP=17\displaystyle {OP}={17}.

a
Find PA\displaystyle PA.

    / – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – check and a new line · * – times · ^ – power, ↓ – end of the power

    / – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

    b
    Find ∠AOP\displaystyle \angle AOP, to two decimal places.

      / – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – check and a new line · * – times · ^ – power, ↓ – end of the power

      / – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

      2

      PA\displaystyle PA and PB\displaystyle PB are tangents to the circle with centre O\displaystyle O, at A\displaystyle A and B\displaystyle B.
      The circle’s radius is 5\displaystyle 5, and ∠APB=60∘\displaystyle {\angle APB}={60^{\circ}}.

      a
      Find ∠AOB\displaystyle \angle AOB.

        / – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – check and a new line · * – times · ^ – power, ↓ – end of the power

        / – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

        b
        Find PA\displaystyle PA. Leave the answer with a square root.

          / – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – check and a new line · * – times · ^ – power, ↓ – end of the power

          / – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

          c
          Find OP\displaystyle OP.

            / – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – check and a new line · * – times · ^ – power, ↓ – end of the power

            / – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

            3

            PA\displaystyle PA and PB\displaystyle PB are tangents to the circle with centre O\displaystyle O, at A\displaystyle A and B\displaystyle B.
            Prove: AB⊥OP\displaystyle AB\perp OP.

            4

            The point A(1,−3)\displaystyle A(1,-3) is on the circle x2+y2=10\displaystyle {x^{2}+y^{2}}={10}.
            Find the equation of the tangent to the circle at A\displaystyle A.

            x2+y2=10\displaystyle x^{2}+y^{2}=10
            A=(1,−3)\displaystyle A=(1,-3)

            5

            The circle (x+4)2+(y−3)2=9\displaystyle {(x+4)^{2}+(y-3)^{2}}={9} is given.
            Which of the axes does the circle touch, and at which point? Explain.

            6

            The circle inscribed in the triangle ABC\displaystyle ABC touches the sides AB\displaystyle AB, BC\displaystyle BC, CA\displaystyle CA at F\displaystyle F, D\displaystyle D, E\displaystyle E.
            AB=13\displaystyle {AB}={13}, BC=14\displaystyle {BC}={14}, CA=15\displaystyle {CA}={15}.

            a
            Find AF\displaystyle AF.

              / – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – check and a new line · * – times · ^ – power, ↓ – end of the power

              / – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

              b
              Find BD\displaystyle BD.

                / – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – check and a new line · * – times · ^ – power, ↓ – end of the power

                / – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

                7

                The legs of a right-angled triangle are 8\displaystyle 8 and 15\displaystyle 15.

                a
                Find the radius of the circle inscribed in the triangle.

                  / – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – check and a new line · * – times · ^ – power, ↓ – end of the power

                  / – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

                  b
                  Find the distance of the inscribed circle’s centre from the vertex of the right angle. Leave the answer with a square root.

                    / – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – check and a new line · * – times · ^ – power, ↓ – end of the power

                    / – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

                    8

                    The four sides of the quadrilateral ABCD\displaystyle ABCD touch the circle with centre O\displaystyle O at K\displaystyle K, L\displaystyle L, M\displaystyle M, N\displaystyle N (the circle is inscribed in the quadrilateral).
                    Prove: AB+CD=AD+BC\displaystyle {AB+CD}={AD+BC}.