Mathematics · Textbook solutions

Circle

Every solved example, exercise, and miscellaneous question — in the order the textbook teaches them. · 31 questions

Properties of a Chord

2 q

Solved Examples

Worked · 2
  1. 6.1 SolvedEx.1
    Radius of a circle is 5 cm. The length of a chord of the circle is 8 cm. Find the distance of the chord from the centre.
  2. 6.1 SolvedEx.2
    Radius of a circle is 20 cm. Distance of a chord from the centre of the circle is 12 cm. Find the length of the chord.

Practice Set 6.1

6 q
  1. Ex 6.1 Q1
    Distance of chord AB from the centre of a circle is 8 cm. Length of the chord AB is 12 cm. Find the diameter of the circle.
  2. Ex 6.1 Q2
    Diameter of a circle is 26 cm and length of a chord of the circle is 24 cm. Find the distance of the chord from the centre.
  3. Ex 6.1 Q3
    Radius of a circle is 34 cm and the distance of the chord from the centre is 30 cm, find the length of the chord.
  4. Ex 6.1 Q4
    Radius of a circle with centre O is 41 units. Length of a chord PQ is 80 units, find the distance of the chord from the centre of the circle.
  5. Ex 6.1 Q5
    In figure 6.9, centre of two circles is O. Chord AB of bigger circle intersects the smaller circle in points P and Q. Show that AP=BQAP = BQ [Fig. 6.9 shows two concentric circles with the common centre O. The chord AB of the bigger circle cuts the smaller circle at P and Q, the four points lying along the chord in the order A, P, Q, B, so that seg PQ is a chord of the smaller circle lying on the same line as chord AB.]
  6. Ex 6.1 Q6
    Prove that, if a diameter of a circle bisects two chords of the circle then those two chords are parallel to each other.

Congruent Chords

1 q

Solved Examples

Worked · 1
  1. 6.2 SolvedEx.1
    In the figure 6.12, O is the centre of the circle and AB=CDAB = CD. If OP=4OP = 4 cm, find the length of OQOQ.

Practice Set 6.2

3 q
  1. Ex 6.2 Q1
    Radius of circle is 10 cm. There are two chords of length 16 cm each. What will be the distance of these chords from the centre of the circle ?
  2. Ex 6.2 Q2
    In a circle with radius 13 cm, two equal chords are at a distance of 5 cm from the centre. Find the lengths of the chords.
  3. Ex 6.2 Q3
    Seg PM and seg PN are congruent chords of a circle with centre C. Show that the ray PC is the bisector of ∠NPM\angle NPM.

Incircle and Circumcircle

2 q

Solved Examples

Worked · 2
  1. 6.3 SolvedEx.1
    Construct △PQR\triangle PQR such that PQ=6PQ = 6 cm, ∠Q=35∘\angle Q = 35^\circ, QR=5.5QR = 5.5 cm. Draw incircle of △PQR\triangle PQR. Draw a rough figure and show all measures in it.
  2. 6.3 SolvedEx.2
    Construct △DEF\triangle DEF such that DE=4.2DE = 4.2 cm, ∠D=60∘\angle D = 60^\circ, ∠E=70∘\angle E = 70^\circ and draw circumcircle of it. Draw rough figure. Write the given measures.

Practice Set 6.3

5 q
  1. Ex 6.3 Q1
    Construct Δ\DeltaABC such that ∠\angleB =100∘= 100^\circ, BC =6.4= 6.4 cm, ∠\angleC =50∘= 50^\circ and construct its incircle.
  2. Ex 6.3 Q2
    Construct Δ\DeltaPQR such that ∠\angleP =70∘= 70^\circ, ∠\angleR =50∘= 50^\circ, QR =7.3= 7.3 cm. and construct its circumcircle.
  3. Ex 6.3 Q3
    Construct Δ\DeltaXYZ such that XY =6.7= 6.7 cm, YZ =5.8= 5.8 cm, XZ =6.9= 6.9 cm. Construct its incircle.
  4. Ex 6.3 Q4
    In Δ\DeltaLMN, LM =7.2= 7.2 cm, ∠\angleM =105∘= 105^\circ, MN =6.4= 6.4 cm, then draw Δ\DeltaLMN and construct its circumcircle.
  5. Ex 6.3 Q5
    Construct Δ\DeltaDEF such that DE == EF =6= 6 cm, ∠\angleF =45∘= 45^\circ and construct its circumcircle.

Problem Set 6

12 q
  1. Choose correct alternative answer and fill in the blanks.
    Prob Q1(i)
    Radius of a circle is 10 cm and distance of a chord from the centre is 6 cm. Hence the length of the chord is .........
    1. A.
      16 cm
    2. B.
      8 cm
    3. C.
      12 cm
    4. D.
      32 cm
  2. Prob Q1(ii)
    The point of concurrence of all angle bisectors of a triangle is called the ......
    1. A.
      centroid
    2. B.
      circumcentre
    3. C.
      incentre
    4. D.
      orthocentre
  3. Prob Q1(iii)
    The circle which passes through all the vertices of a triangle is called .....
    1. A.
      circumcircle
    2. B.
      incircle
    3. C.
      congruent circle
    4. D.
      concentric circle
  4. Prob Q1(iv)
    Length of a chord of a circle is 24 cm. If distance of the chord from the centre is 5 cm, then the radius of that circle is ....
    1. A.
      12 cm
    2. B.
      13 cm
    3. C.
      14 cm
    4. D.
      15 cm
  5. Prob Q1(v)
    The length of the longest chord of the circle with radius 2.9 cm is .....
    1. A.
      3.5 cm
    2. B.
      7 cm
    3. C.
      10 cm
    4. D.
      5.8 cm
  6. Prob Q1(vi)
    Radius of a circle with centre O is 4 cm. If l(OP)=4.2l(\text{OP}) = 4.2 cm, say where point P will lie.
    1. A.
      on the centre
    2. B.
      Inside the circle
    3. C.
      outside the circle
    4. D.
      on the circle
  7. Prob Q1(vii)
    The lengths of parallel chords which are on opposite sides of the centre of a circle are 6 cm and 8 cm. If radius of the circle is 5 cm, then the distance between these chords is .....
    1. A.
      2 cm
    2. B.
      1 cm
    3. C.
      8 cm
    4. D.
      7 cm
  8. Prob Q2
    Construct incircle and circumcircle of an equilateral Δ\DeltaDSP with side 7.5 cm. Measure the radii of both the circles and find the ratio of radius of circumcircle to the radius of incircle.
  9. Prob Q3
    Construct Δ\DeltaNTS where NT =5.7= 5.7 cm, TS =7.5= 7.5 cm and ∠\angleNTS =110∘= 110^\circ and draw incircle and circumcircle of it.
  10. Prob Q4
    In the figure 6.19, C is the centre of the circle. seg QT is a diameter, CT =13= 13, CP =5= 5, find the length of chord RS. (In the figure the diameter QT is drawn vertically with Q at the top and T at the bottom, and the chord RS is drawn horizontally with R on the left and S on the right. RS cuts the diameter at the point P, where a right angle is marked, so seg QT ⊥\perp chord RS at P; P lies between Q and the centre C.)
  11. Prob Q5
    In the figure 6.20, P is the centre of the circle. Chord AB and chord CD intersect on the diameter at the point E. If ∠\angleAEP ≅\cong ∠\angleDEP then prove that AB == CD. (In the figure the diameter through P is drawn vertically; chord AB runs from B at the upper left to A at the right, and chord CD runs from C at the upper right to D at the left. The two chords cross at E, which lies on that diameter above the centre P, and the two angles ∠\angleAEP and ∠\angleDEP at E are marked with arcs.)
  12. Prob Q6
    In the figure 6.21, CD is a diameter of the circle with centre O. Diameter CD is perpendicular to chord AB at point E. Show that Δ\DeltaABC is an isosceles triangle. (In the figure the diameter CD is drawn vertically with C at the top and D at the bottom, and the chord AB is drawn horizontally with A on the left and B on the right. CD meets AB at E, where the right angle is marked, and seg CA and seg CB are drawn so that A, B and C are the vertices of the triangle.)