Figure 064 : Tangent and circles
Question and Answer

    Read Symbol defintion

  • Q01 | - Diagrams : Tangents and circles
  • Q02 | - Prove that two tangents at point P to circle are equal.
  • Q03 | - Draw tangent on given point P on circle
  • Q04 | - Draw tangents PQ and PR to a circle so that angle QPR = 30.
  • Q05 | - How to draw tangents to a circle from a point outside the circle ?
  • Q06 | - Circle is inscribed a quadrilateral
  • Q07 | - Facts and theories
  • Q08 | - Exercises
Answers


Q01. Diagrams : Tangents and circles


Top left diagram
  • Two points are fixed A and B. Point P moves so that angle APB = 90 degrees.
  • What is locus of point P ?
  • If PA^2 + PB^2 = AB, what is the locus of P ?
Top mid diagram
  • One point O is fixed. Point P moves so that OP = constant.
  • What is the locus ?
  • It is a circle. the equations is (xP-xO)^2 + (yP-yO)^2 = OP^2.
Top right diagram
  • From point P draw two tangents which touch the circle at Q and R.
  • Then we have the following theory.
    • Two tangents PQ = PR.
    • OQ is perpendicular to PQ and OR perpendicular to PR.
    • Angle QOR + angle QPR = 180 degrees.
Bottom left diagram
  • Excenter and excircle of a triangle
Bottom mid diagram
  • Incenter and incircle of a triangle
Bottom right diagram
  • Excenter and excircle of a triangle.
Go to Begin

Q02. Prove that two tangents at point P to circle are equal

Proof : Use the diagram constructed in Q1
  • Angle PQO = 90 and angle PRO = 90.
  • QO = RO and OP as common sides of right angle triangle PQO and PRO.
  • Hence triangle PQO is congruent to triangle PRO.
  • Hence PQ = PR.
Theory used to prove that two triangles are congruent
  • Condition 1 : SSS means three sides of two triangle are equal.
  • Condition 2 : SAS means sides include angle of two triangle are same.
  • Condition 3 : ASA means two angles and include side of two triangles are same.
Q. Whta is the condition used to prove two right angle triangles ?
  • Since OQ = OR and OP is common.
  • Hence PQ = PR = Sqr(OQ^2 + OP^2).
  • Hence the SSS condtion is satisfied by triangles POQ and PRO.
  • We also say that
    • PQ = PR, Angle PQO = angle PRO = 90, OQ = OR.
    • Hence the condtion is SAS.

Go to Begin

Q03. Draw tangent on given point on circle

Construction
  • Draw a circle with center O and radius r.
  • Draw a point P on the circle.
  • Extend line OP to point Q so that OP = PQ.
  • Use O as cneter and draw a circle with radius a where a is greater than r.
  • Use Q as cneter and draw a circle with same radius a.
  • The two circles with radius a are intersected at R and T
  • Join R and T.
Proof
  • Then RT is perpendicular to PO.
  • Since PO is radius and P is on the circle, hence RT is the reuired tangent.

Go to Begin

Q04. Draw tangent PQ and PR to a circle so that angle QPR = 30

Construction
  • Draw a circle with center O.
  • Draw a diameter RT.
  • Draw a line OQ so that Q on the circle and OP makes angel 30 degree with RT.
  • Join PQ and PR which are the required tangents.
Proof
  • By theory : Angle QPR + angle QOR = pi.
  • By construction : Angle QOR + angle QOT = pi.
  • Since Angle QOT = 30 degrees.
  • Hence angle QPR = 30 degrees.

Go to Begin

Q05. Draw two tangents to a circle from point P ourside the circle

Construction.
  • Draw a circle and let center be O.
  • Draw a point P at outside of the circle.
  • Join point P and O.
  • Use PO as diameter to draw a another circle.
  • Two circles intersect at point Q and R.
  • The PQ and PR are the required tangents as shown in top right diagram.
Proof
  • Join point Q and center O.
  • Join point R and center O.
  • By theory : angle PQO = 90 degrees and angle PRO = 90 degrees.
  • Since Tangent PQ is perpendicular to radius QO.
  • Hence PQ is the tangent to the circle.
  • Similarly Tangent PR is perpendicular to radius RO.
  • Hence PR is the tangent to the circle.
Theories related to this construction
  • 1. The tangent at point Q on circle is perpendicular to the radius QO.
  • 2. Use diameter AB make a triangle ABC inscribed in a circle and angle ACB = 90 degrees.
  • 3. Tangents PQ and PR are equal.
  • 4. The angles QPR + QOR = 180 degrees.

Go to Begin

Q06. Circle is inscribed a quadrilateral

Construction
  • Draw a circle.
  • Draw quadrilateral ABCD and their sides tangent to the circle.
  • Let AB tangent to circle at P.
  • Let BC tangent to circle at Q.
  • Let CD tangent to circle at R.
  • Let CA tangent to circle at T.
  • prove that AB + CD = AD + BC.
Proof
  • Two tangents equal : AP = AT, BP = BQ, CQ = CR, DR = DT.
  • Hence AB + CD
  • = AP + PB + CR + RD
  • = AT + BQ + CQ + DT
  • = AT + DT + BQ + CQ
  • = AD + BC.

Go to Begin

Q07. Facts and theories

  • 1. The tangent at point Q on circle is perpendicular to the radius QO.
  • 2. Triangle ABC inscribed in a circle
    • Let AB = diameter
    • Then angle ACB = 90 degrees.
  • 3. Tangents PQ and PR are equal.
  • 4. The angles QPR + QOR = 180 degrees.

Go to Begin

Q08. Exercises

  • 1. What is tangent to a circle ?
  • 2. What is length of a tangent from P to a circle at point T ?
  • 3. How to draw a line which bisect a line AB ?
  • 4. Find locus of point P, if Two points A and B are fixed and angle APQ = 90 degrees.
Go to Begin
Show Room of MD2002 Contact Dr. Shih Math Examples Room
Copyright © Dr. K. G. Shih. Nova Scotia, Canada.
1