Mathematics · Textbook solutions

Basic Concepts in Geometry

Every solved example, exercise, and miscellaneous question — in the order the textbook teaches them. · 86 questions

Co-ordinates, Distance and Betweenness

3 q

Solved Examples

Worked · 3
  1. 1.1 SolvedEx.1
    On a number line, points A, B and C are such that d(A,B)=5d(A, B) = 5, d(B,C)=11d(B, C) = 11 and d(A,C)=6d(A, C) = 6. Which of the points is between the other two?
  2. 1.1 SolvedEx.2
    U, V and A are three cities on a straight road. The distance between U and A is 215 km, between V and A is 140 km and between U and V is 75 km. Which of them is between the other two?
  3. 1.1 SolvedEx.3
    The co-ordinate of point A on a number line is 5. Find the co-ordinates of points on the same number line which are 13 units away from A.

Practice Set 1.1

26 q
  1. Find the distances with the help of the number line given below (Fig. 1.5), on which the points Q, P, K, J, H, O, A, B, C, D and E have the co-ordinates −5-5, −4-4, −3-3, −2-2, −1-1, 0, 1, 2, 3, 4 and 5 respectively.
    Ex 1.1 Q1(i)
    From the number line in Fig. 1.5, in which the co-ordinate of B is 2 and that of E is 5, find d(B,E)d(B, E).
  2. Ex 1.1 Q1(ii)
    From the number line in Fig. 1.5, in which the co-ordinate of J is −2-2 and that of A is 1, find d(J,A)d(J, A).
  3. Ex 1.1 Q1(iii)
    From the number line in Fig. 1.5, in which the co-ordinate of P is −4-4 and that of C is 3, find d(P,C)d(P, C).
  4. Ex 1.1 Q1(iv)
    From the number line in Fig. 1.5, in which the co-ordinate of J is −2-2 and that of H is −1-1, find d(J,H)d(J, H).
  5. Ex 1.1 Q1(v)
    From the number line in Fig. 1.5, in which the co-ordinate of K is −3-3 and that of O is 0, find d(K,O)d(K, O).
  6. Ex 1.1 Q1(vi)
    From the number line in Fig. 1.5, in which the co-ordinate of O is 0 and that of E is 5, find d(O,E)d(O, E).
  7. Ex 1.1 Q1(vii)
    From the number line in Fig. 1.5, in which the co-ordinate of P is −4-4 and that of J is −2-2, find d(P,J)d(P, J).
  8. Ex 1.1 Q1(viii)
    From the number line in Fig. 1.5, in which the co-ordinate of Q is −5-5 and that of B is 2, find d(Q,B)d(Q, B).
  9. If the co-ordinate of A is xx and that of B is yy, find d(A,B)d(A, B).
    Ex 1.1 Q2(i)
    x=1x = 1, y=7y = 7
  10. Ex 1.1 Q2(ii)
    x=6x = 6, y=−2y = -2
  11. Ex 1.1 Q2(iii)
    x=−3x = -3, y=7y = 7
  12. Ex 1.1 Q2(iv)
    x=−4x = -4, y=−5y = -5
  13. Ex 1.1 Q2(v)
    x=−3x = -3, y=−6y = -6
  14. Ex 1.1 Q2(vi)
    x=4x = 4, y=−8y = -8
  15. From the information given below, find which of the point is between the other two. If the points are not collinear, state so.
    Ex 1.1 Q3(i)
    d(P,R)=7d(P, R) = 7, d(P,Q)=10d(P, Q) = 10, d(Q,R)=3d(Q, R) = 3
  16. Ex 1.1 Q3(ii)
    d(R,S)=8d(R, S) = 8, d(S,T)=6d(S, T) = 6, d(R,T)=4d(R, T) = 4
  17. Ex 1.1 Q3(iii)
    d(A,B)=16d(A, B) = 16, d(C,A)=9d(C, A) = 9, d(B,C)=7d(B, C) = 7
  18. Ex 1.1 Q3(iv)
    d(L,M)=11d(L, M) = 11, d(M,N)=12d(M, N) = 12, d(N,L)=8d(N, L) = 8
  19. Ex 1.1 Q3(v)
    d(X,Y)=15d(X, Y) = 15, d(Y,Z)=7d(Y, Z) = 7, d(X,Z)=8d(X, Z) = 8
  20. Ex 1.1 Q3(vi)
    d(D,E)=5d(D, E) = 5, d(E,F)=8d(E, F) = 8, d(D,F)=6d(D, F) = 6
  21. Ex 1.1 Q4
    On a number line, points A, B and C are such that d(A,C)=10d(A, C) = 10, d(C,B)=8d(C, B) = 8. Find d(A,B)d(A, B) considering all possibilities.
  22. Ex 1.1 Q5
    Points X, Y, Z are collinear such that d(X,Y)=17d(X, Y) = 17, d(Y,Z)=8d(Y, Z) = 8, find d(X,Z)d(X, Z).
  23. Sketch proper figure and write the answers of the following questions.
    Ex 1.1 Q6(i)
    If A−B−CA-B-C and l(AC)=11l(AC) = 11, l(BC)=6.5l(BC) = 6.5, then l(AB)=l(AB) = ?
  24. Ex 1.1 Q6(ii)
    If R−S−TR-S-T and l(ST)=3.7l(ST) = 3.7, l(RS)=2.5l(RS) = 2.5, then l(RT)=l(RT) = ?
  25. Ex 1.1 Q6(iii)
    If X−Y−ZX-Y-Z and l(XZ)=37l(XZ) = 3\sqrt{7}, l(XY)=7l(XY) = \sqrt{7}, then l(YZ)=l(YZ) = ?
  26. Ex 1.1 Q7
    Which figure is formed by three non-collinear points?

Practice Set 1.2

16 q
  1. The following table shows points on a number line and their co-ordinates. Decide whether the pair of segments given below the table are congruent or not.
    PointABCDE
    Co-ordinate−3-35522−7-799
    Ex 1.2 Q1(i)
    seg DE and seg AB
  2. Ex 1.2 Q1(ii)
    seg BC and seg AD
  3. Ex 1.2 Q1(iii)
    seg BE and seg AD
  4. Ex 1.2 Q2
    Point M is the midpoint of seg AB. If AB=8AB = 8 then find the length of AM.
  5. Ex 1.2 Q3
    Point P is the midpoint of seg CD. If CP=2.5CP = 2.5, find l(CD)l(CD).
  6. Ex 1.2 Q4
    If AB=5AB = 5 cm, BP=2BP = 2 cm and AP=3.4AP = 3.4 cm, compare the segments.
  7. Write the answers to the following questions with reference to figure 1.13. In figure 1.13, the points T, S, R, P and Q lie on one line, in that order from left to right.
    Ex 1.2 Q5(i)
    In figure 1.13, points T, S, R, P, Q lie on one line in that order (left to right). Write the name of the opposite ray of ray RP.
  8. Ex 1.2 Q5(ii)
    In figure 1.13, points T, S, R, P, Q lie on one line in that order (left to right). Write the intersection set of ray PQ and ray RP.
  9. Ex 1.2 Q5(iii)
    In figure 1.13, points T, S, R, P, Q lie on one line in that order (left to right). Write the union set of seg PQ and seg QR.
  10. Ex 1.2 Q5(iv)
    In figure 1.13, points T, S, R, P, Q lie on one line in that order (left to right). State the rays of which seg QR is a subset.
  11. Ex 1.2 Q5(v)
    In figure 1.13, points T, S, R, P, Q lie on one line in that order (left to right). Write the pair of opposite rays with common end point R.
  12. Ex 1.2 Q5(vi)
    In figure 1.13, points T, S, R, P, Q lie on one line in that order (left to right). Write any two rays with common end point S.
  13. Ex 1.2 Q5(vii)
    In figure 1.13, points T, S, R, P, Q lie on one line in that order (left to right). Write the intersection set of ray SP and ray ST.
  14. Answer the questions with the help of figure 1.14. Figure 1.14 shows a number line on which the marked points have these co-ordinates: R −6-6, U −5-5, Q −4-4, L −3-3, P −2-2, A 00, B 22, C 44, V 55, D 66.
    Ex 1.2 Q6(i)
    On the number line of figure 1.14 the points have co-ordinates R −6-6, U −5-5, Q −4-4, L −3-3, P −2-2, A 00, B 22, C 44, V 55, D 66. State the points which are equidistant from point B.
  15. Ex 1.2 Q6(ii)
    On the number line of figure 1.14 the points have co-ordinates R −6-6, U −5-5, Q −4-4, L −3-3, P −2-2, A 00, B 22, C 44, V 55, D 66. Write a pair of points equidistant from point Q.
  16. Ex 1.2 Q6(iii)
    On the number line of figure 1.14 the points have co-ordinates R −6-6, U −5-5, Q −4-4, L −3-3, P −2-2, A 00, B 22, C 44, V 55, D 66. Find d(U,V)d(U,V), d(P,C)d(P,C), d(V,B)d(V,B), d(U,L)d(U,L).

Conditional Statements and Proof

2 q

Solved Examples

Worked · 2
  1. 1.3 SolvedEx.1
    Theorem : The opposite angles formed by two intersecting lines are of equal measures. Line AB and line CD intersect at point O such that A - O - B, C - O - D. Prove that (i) ∠AOC=∠BOD\angle AOC = \angle BOD (ii) ∠BOC=∠AOD\angle BOC = \angle AOD.
  2. 1.3 SolvedEx.2
    Statement : A prime number greater than 2 is odd. Write this statement in conditional form and prove it by an indirect proof.

Practice Set 1.3

6 q
  1. Write the following statements in 'if-then' form.
    Ex 1.3 Q1(i)
    The opposite angles of a parallelogram are congruent.
  2. Ex 1.3 Q1(ii)
    The diagonals of a rectangle are congruent.
  3. Ex 1.3 Q1(iii)
    In an isosceles triangle, the segment joining the vertex and the mid point of the base is perpendicular to the base.
  4. Write converses of the following statements.
    Ex 1.3 Q2(i)
    The alternate angles formed by two parallel lines and their transversal are congruent.
  5. Ex 1.3 Q2(ii)
    If a pair of the interior angles made by a transversal of two lines are supplementary then the lines are parallel.
  6. Ex 1.3 Q2(iii)
    The diagonals of a rectangle are congruent.

Problem Set 1

33 q
  1. Select the correct alternative from the answers of the questions given below.
    Prob Q1(i)
    How many mid points does a segment have ?
    1. A.
      only one
    2. B.
      two
    3. C.
      three
    4. D.
      many
  2. Prob Q1(ii)
    How many points are there in the intersection of two distinct lines ?
    1. A.
      infinite
    2. B.
      two
    3. C.
      one
    4. D.
      not a single
  3. Prob Q1(iii)
    How many lines are determined by three distinct points ?
    1. A.
      two
    2. B.
      three
    3. C.
      one or three
    4. D.
      six
  4. Prob Q1(iv)
    Find d(A,B)d(A, B), if co-ordinates of A and B are −2-2 and 55 respectively.
    1. A.
      −2-2
    2. B.
      55
    3. C.
      77
    4. D.
      33
  5. Prob Q1(v)
    If P-Q-RP\text{-}Q\text{-}R and d(P,Q)=2d(P,Q) = 2, d(P,R)=10d(P,R) = 10, then find d(Q,R)d(Q,R).
    1. A.
      1212
    2. B.
      88
    3. C.
      96\sqrt{96}
    4. D.
      2020
  6. On a number line, co-ordinates of P, Q, R are 33, −5-5 and 66 respectively. State with reason whether the following statements are true or false.
    Prob Q2(i)
    d(P,Q)+d(Q,R)=d(P,R)d(P,Q) + d(Q,R) = d(P,R)
  7. Prob Q2(ii)
    d(P,R)+d(R,Q)=d(P,Q)d(P,R) + d(R,Q) = d(P,Q)
  8. Prob Q2(iii)
    d(R,P)+d(P,Q)=d(R,Q)d(R,P) + d(P,Q) = d(R,Q)
  9. Prob Q2(iv)
    d(P,Q)−d(P,R)=d(Q,R)d(P,Q) - d(P,R) = d(Q,R)
  10. Co-ordinates of some pairs of points are given below. Hence find the distance between each pair.
    Prob Q3(i)
    3, 63,\ 6
  11. Prob Q3(ii)
    −9, −1-9,\ -1
  12. Prob Q3(iii)
    −4, 5-4,\ 5
  13. Prob Q3(iv)
    0, −20,\ -2
  14. Prob Q3(v)
    x+3, x−3x + 3,\ x - 3
  15. Prob Q3(vi)
    −25, −47-25,\ -47
  16. Prob Q3(vii)
    80, −8580,\ -85
  17. Prob Q4
    Co-ordinate of point P on a number line is −7-7. Find the co-ordinates of points on the number line which are at a distance of 8 units from point P.
  18. Answer the following questions.
    Prob Q5(i)
    If A-B-CA\text{-}B\text{-}C and d(A,C)=17d(A,C) = 17, d(B,C)=6.5d(B,C) = 6.5 then d(A,B)=?d(A,B) = ?
  19. Prob Q5(ii)
    If P-Q-RP\text{-}Q\text{-}R and d(P,Q)=3.4d(P,Q) = 3.4, d(Q,R)=5.7d(Q,R) = 5.7 then d(P,R)=?d(P,R) = ?
  20. Prob Q6
    Co-ordinate of point A on a number line is 1. What are the co-ordinates of points on the number line which are at a distance of 7 units from A ?
  21. Write the following statements in conditional form.
    Prob Q7(i)
    Every rhombus is a square.
  22. Prob Q7(ii)
    Angles in a linear pair are supplementary.
  23. Prob Q7(iii)
    A triangle is a figure formed by three segments.
  24. Prob Q7(iv)
    A number having only two divisors is called a prime number.
  25. Write the converse of each of the following statements.
    Prob Q8(i)
    If the sum of measures of angles in a figure is 180∘180^\circ, then the figure is a triangle.
  26. Prob Q8(ii)
    If the sum of measures of two angles is 90∘90^\circ then they are complement of each other.
  27. Prob Q8(iii)
    If the corresponding angles formed by a transversal of two lines are congruent then the two lines are parallel.
  28. Prob Q8(iv)
    If the sum of the digits of a number is divisible by 3 then the number is divisible by 3.
  29. Write the antecedent (given part) and the consequent (part to be proved) in the following statements.
    Prob Q9(i)
    If all sides of a triangle are congruent then its all angles are congruent.
  30. Prob Q9(ii)
    The diagonals of a parallelogram bisect each other.
  31. Q.10* (starred — optional / harder). Draw a labelled figure showing information in each of the following statements and write the antecedent and the consequent.
    Prob Q10*(i)
    Two equilateral triangles are similar.
  32. Prob Q10*(ii)
    If angles in a linear pair are congruent then each of them is a right angle.
  33. Prob Q10*(iii)
    If the altitudes drawn on two sides of a triangle are congruent then those two sides are congruent.