## Chapter 1 – Basic Concepts in Geometry

**Practice set 1.1**

**1. Find the distances with the help of the number line given below.**

**(i) d(B,E) **

**Solution:**

The coordinate of point B is 2.

The coordinate of point E is 5.

5 > 2

d(B, E) = Greater co–ordinate – smaller coordinate

d(B, E) = 5 – 2

d(B, E) = 3

**Ans:** d(B, E) = 3 units

**(ii) d(J, A) **

**Solution:**

The coordinate of point J is –2.

The coordinate of point A is 1.

1 > – 2

d(J, A) = Greater co–ordinate – smaller co–ordinate

d(J, A) = 1 – (–2)

d(J, A) = 1 + 2

d(J, A) = 3

**Ans:** d(J, A) = 3 units

**(iii) d(P, C) **

**Solution:**

The co–ordinate of point P is – 4.

The co–ordinate of point C is 3.

3 > – 4

d(P, C) = Greater co–ordinate – smaller co–ordinate

d(P, C) = 3 – (– 4)

d(P, C) = 3 + 4

d(P, C) = 7

**Ans:** d(P, C) = 7 units

**(iv) d(J, H) **

**Solution:**

The coordinate of point J is –2.

The coordinate of point H is –1.

– 1 > – 2

d(J, H) = Greater coordinate – smaller coordinate

d(J, H) = – 1 – (– 2)

d(J, H) = – 1 + 2

d(J, H) = 1

**Ans:** d(J, H) = 1 unit

**(v) d(K, O) **

**Solution:**

The coordinate of point K is –3.

The coordinate of point O is 0.

0 > –3

d(K, O) = Greater coordinate – smaller coordinate

d(K, O) = 0 – (–3)

d(K, O) = 0 + 3

d(K, O) = 3

**Ans:** d(K, O) = 3 units

**(vi) d(O, E) **

**Solution:**

The coordinate of point O is 0.

The coordinate of point E is 5.

5 > 0

d(O, E) = Greater coordinate – smaller coordinate

d(O, E) = 5 – 0

d(O, E) = 5

**Ans:** d(O, E) = 5 units

**(vii) d(P, J) **

**Solution:**

The coordinate of point P is – 4.

The coordinate of point J is – 2.

– 2 > – 4

d(P, J) = Greater coordinate – smaller coordinate

d(P, J) = – 2 – (– 4)

d(P, J) = – 2 + 4

d(P, J) = 2

**Ans:** d(P, J) = 2 units

**(viii) d(Q, B)**

**Solution:**

The coordinate of point Q is – 5.

The coordinate of point B is 2.

2 > – 5

d(Q, B) = Greater coordinate – smaller coordinate

d(Q, B) = 2 – (– 5)

d(Q, B) = 2 + 5

d(Q, B) = 7

**Ans:** d(Q, B) = 7 units

**2. If the coordinate of A is x and that of B is y, find d(A, B). **

**(i) x = 1, y = 7 **

**Solution:**

The coordinate of point A is 1.

The coordinate of point B is 7.

7 > 1

d(A, B) = Greater coordinate – smaller coordinate

d(A, B) = 7 – 1

d(A, B) = 6

**Ans:** d(A, B) = 6 units

**(ii) x = 6, y = – 2 **

**Solution:**

The coordinate of point A is 6.

The coordinate of point B is – 2.

6 > – 2

d(A, B) = Greater coordinate – smaller coordinate

d(A, B) = 6 – (– 2)

d(A, B) = 6 + 2

d(A, B) = 8

**Ans:** d(A, B) = 8 units

**(iii) x = – 3, y = 7**

**Solution:**

The coordinate of point A is – 3.

The coordinate of point B is 7.

7 > – 3

d(A, B) = Greater coordinate – smaller coordinate

d(A, B) = 7 – (– 3)

d(A, B) = 7 + 3

d(A, B) = 10

**Ans:** d(A, B) = 10 units

**(iv) x = – 4, y = – 5 **

**Solution:**

The coordinate of point A is –4.

The coordinate of point B is –5.

– 4 > – 5

d(A, B) = Greater coordinate – smaller coordinate

d(A, B) = – 4 – (– 5)

d(A, B) = – 4 + 5

d(A, B) = 1

**Ans:** d(A, B) = 1 unit

**(v) x = – 3, y = – 6 **

**Solution:**

The coordinate of point A is –3.

The coordinate of point B is – 6.

– 3 > – 6

d(A, B) = Greater coordinate – smaller coordinate

d(A, B) = – 3 – (– 6)

d(A, B) = – 3 + 6

d(A, B) = 3

**Ans:** d(A, B) = 3 units

**(vi) x = 4, y = – 8**

**Solution:**

The coordinate of point A is 4.

The coordinate of point B is – 8.

4 > – 8

d(A, B) = Greater coordinate – smaller coordinate

d(A, B) = 4 – (– 8)

d(A, B) = 4 + 8

d(A, B) = 12

**Ans:** d(A, B) = 12 units

**3. From the information given below, find which of the points is between the other two. If the points are not collinear, state so. **

**(i) d(P, R) = 7, d(P, Q) = 10, d(Q, R) = 3**

**Solution:**

d(P, Q) = 10 …(i)

d(P, R) + d(Q, R) = 7 + 3

∴ d(P, R) + d(Q, R) = 10 …(ii)

∴ d(P, Q) = d(P, R) + d(Q, R) …[*From (i) and (ii)*]

∴ Points P, Q and R are collinear points.

∴ Relation of betweenness exists, P – R – Q.

**(ii) d(R, S) = 8, d(S, T) = 6, d(R, T) = 4**

**Solution:**

d(R, S) = 8 …(i)

d(S, T) + d(R, T) = 6 + 4

∴ d(S, T) + d(R, T) = 10 …(ii)

∴ d(R, S) ≠ d(S, T) + d(R, T) …[*From (i) and (ii)*]

∴ Points R, S and T are non-collinear points.

∴ Relation of betweenness does not exist.

**(iii) d(A, B) = 16, d(C, A) = 9, d(B, C) = 7**

**Solution:**

d(A, B) = 16 ….. (i)

d(C, A) + d(B, C) = 9 + 7

∴ d(C, A) + d(B, C) = 16 ….. (ii)

∴ d(A, B) = d(C, A) + d(B, C) …[*From (i) and (ii)*]

∴ Points A, B and C are collinear points.

∴ Relation of betweenness exists, A – C – B.

**(iv) d(L, M) = 11, d(M, N) = 12, d(N, L) = 8**

**Solution:**

d(M, N) = 12 …(i)

d(L, M) + d(N, L) = 11 + 8

∴ d(L, M) + d(N, L) = 19 …(ii)

∴ d(M, N) ≠ d(L, M) + d(N, L) …[*From (i) and (ii)*]

∴ Points L, M and N are non-collinear points.

∴ Relation of betweenness does not exist.

**(v) d(X, Y) = 15, d(Y, Z) = 7, d(X, Z) = 8**

**Solution:**

d(X, Y) = 15 …(i)

d(Y, Z) + d(X, Z) = 7 + 8

∴ d(Y, Z) + d(X, Z) = 15 ….. (ii)

∴ d(X, Y) = d(Y, Z) + d(X, Z) …[*From (i) and (ii)*]

∴ Points X, Y and Z are collinear points.

∴ Relation of betweenness exists, X – Z – Y.

**(vi) d(D, E) = 5, d(E, F) = 8, d(D, F) = 6**

**Solution:**

d(E, F) = 8 …(i)

d(D, E) + d(D, F) = 5 + 6

∴ d(D, E) + d(D, F) = 11 …(ii)

∴ d(E, F) ≠ d(D, E) + d(D, F) …[*From (i) and (ii)*]

∴ Points D, E and F are non-collinear points.

∴ Relation of betweenness does not exist.

**4. On a number line, points A, B and C are such that d(A, C) = 10, d(C, B) = 8. Find d(A, B) considering all possibilities. **

**Solution:**

Case (i):

Consider A – B – C

∴ d(A, C) = d(A, B) + d(B, C)

∴ 10 = d(A, B) + 8

∴ 10 – 8 = d(A, B)

∴ d(A, B) = 2 units

Case (ii):

Consider A – C – B

∴ d(A, B) = d(A, C) + d(C, B)

∴ d(A, B) = 10 + 8

∴ d(A, B) = 18 units

Case (iii):

Consider B – A – C

∴ d(A, B) + d(A, C) = d(C, B)

∴ d(A, B) + 10 = 8

∴ d(A, B) = 8 – 10

∴ d(A, B) = – 2

But the distance between two points cannot be negative.

∴ B – A – C is not possible.

∴ d(A, B) = 2 units or d(A, B) = 18 units

**Ans:** d(A, B) can be 2 units or 18 units.

**5. Points X, Y, Z are collinear such that d(X, Y) = 17, d(Y, Z) = 8, find d(X, Z). **

**Solution:**

Case (i):

Consider X – Y – Z

∴ d(X, Z) = d(X, Y) + d(Y, Z)

∴ d(X, Z) = 17 + 8

∴ d(X, Z) = 25 units

Case (ii):

Consider X – Z – Y

∴ d(X, Y) = d(X, Z) + d(Z, Y)

∴ 17 = d(X, Z) + 8

∴ 17 – 8 = d(X, Z)

∴ d(X, Z) = 9 units

Case (iii):

Consider Y – X – Z

∴ d(X, Y) + d(X, Z) = d(Y, Z)

∴ 17 + d(X, Z) = 8

∴ d(X, Z) = – 17 + 8

∴ d(X, Z) = – 9 units

But, distance between two points cannot be negative.

Y – X – Z is not possible.

∴ d(X, Z) = 25 units or d(X, Z) = 9 units

**Ans:** d(X, Z) can be 9 units or 25 units.

**6. Sketch a proper figure and write the answers of the following questions. **

**(i) If A – B – C and l(AC) = 11, l(BC) = 6.5, then l(AB) = ?**

**Solution:**

A – B – C …[*Given*]

l(AC) = l(AB) + l(BC)

11 = l(AB) + 6.5

11 – 6.5 = l(AB)

l(AB) = 4.5 units

**Ans:** l(AB) = 4.5 units

**Figure:**

**(ii) If R – S – T and l(ST) = 3.7, l(RS) = 2.5, then l(RT) = ?**

**Solution:**

R – S – T …[*Given*]

l(RT) = l(RS) + l(ST)

l(RT) = 2.5 + 3.7

l(RT) = 6.2 units

**Ans:** l(RT) = 6.2 units

**Figure:**

**(iii) If X – Y – Z and l(XZ) = 3 \(\sqrt {7}\), l(XY) = \(\sqrt {7}\), then l(YZ) = ?**

**Solution:**

X – Y – Z …[*Given*]

l(XZ) = l(XY) + l(YZ)

3 \(\sqrt {7}\) = \(\sqrt {7}\) + l(YZ)

3 \(\sqrt {7}\) – \(\sqrt {7}\) = l(YZ)

l(YZ) = 2 \(\sqrt {7}\) units

**Ans: l(YZ) = 2 \(\sqrt {7}\) units**

**Figure:**

**7. Which figure is formed by three non-collinear points ?**

**Ans:** Three non-collinear points join to form a triangle.

**Figure:**

**Practice set 1.2**

**1. The following table shows points on a number line and their coordinates. Decide whether the pair of segments given below the table are congruent or not.**

**(i) seg DE and seg AB **

**Solution:**

The coordinate of point D is – 7.

The coordinate of point E is 9.

9 > – 7

d(D, E) = Greater coordinate – Smaller co-ordinate

d(D, E) = 9 – (– 7)

d(D, E) = 9 + 7

d(D, E) = 16

∴ l(DE) = 16 units ….(i)

The coordinate of point A is – 3.

The coordinate of point B is 5.

5 > – 3

d(A, B) = Greater coordinate – Smaller coordinate

d(A, B) = 5 – (– 3)

d(A, B) = 5 + 3

d(A, B) = 8

l(AB) = 8 units …(ii)

∴ l(DE) ≠ l(AB) …[*From (i) and (ii)*]

**Ans:** Seg DE is not congruent to seg AB.

**(ii) seg BC and seg AD**

**Solution:**

The coordinate of point B is 5.

The coordinate of point C is 2.

5 > 2

d(B, C) = Greater coordinate – Smaller coordinate

d(B, C) = 5 – 2

d(B, C) = 3

l(BC) = 3 units …(i)

The coordinate of point A is – 3.

The coordinate of point D is – 7.

– 3 > – 7

d(A, D) = Greater coordinate – Smaller coordinate

d(A, D) = – 3 – (– 7)

d(A, D) = – 3 + 7

d(A, D) = 4

l(AD) = 4 units …(ii)

∴ l(BC) ≠ l(AD) …[*From (i) and (ii)*]

**Ans:** Seg BC is not congruent to seg AD.

**(iii) seg BE and seg AD**

**Solution:**

The coordinate of point B is 5.

The coordinate of point E is 9.

9 > 5

d(B, E) = Greater coordinate – Smaller coordinate

d(B, E) = 9 – 5

d(B, E) = 4

l(BE) = 4 units …(i)

The coordinate of point A is – 3.

The coordinate of point D is – 7.

– 3 > – 7

d(A, D) = Greater coordinate – Smaller coordinate

d(A, D) = – 3 – (– 7)

d(A, D) = – 3 + 7

d(A, D) = 4

l(AD) = 4 units …(ii)

∴ l(BE) = l(AD) …[*From (i) and (ii)*]

**Ans:** seg BE ≅ seg AD

**2. Point M is the midpoint of seg AB. If AB = 8 then find the length of AM.**

**Solution:**

M is the midpoint of seg AB …[*Given*]

l(AM) = \(\large \frac {1}{2}\) l(AB)

l(AM) = \(\large \frac {1}{2}\) × 8

l(AM) = 4 units

**Ans:** l(AM) = 4 units

**3. Point P is the midpoint of seg CD. If CP = 2.5, find l(CD).**

**Solution:**

Point P is the midpoint of seg CD …[*Given*]

l(CP) = \(\large \frac {1}{2}\) l(CD)

2.5 = \(\large \frac {1}{2}\) l(CD)

l(CD) = 2.5 × 2

l(CD) = 5 units

**Ans:** l(CD) = 5 units

**4. If AB = 5 cm, BP = 2 cm and AP = 3.4 cm, compare the segments.**

**Solution:**

AB = 5 cm, BP = 2 cm, AP = 3.4 cm

5 > 3.4 > 2

∴ l(AB) > l(AP) > l(BP)

**5. Write the answers to the following questions with reference to figure 1.13.**

**(i) Write the name of the opposite ray of ray RP.**

**Ans:** Ray RS is opposite of ray RP.

**(ii) Write the intersection set of ray PQ and ray RP.**

**Ans:** Intersection set of ray PQ and ray RP is ray PQ.

**(iii) Write the union set of seg PQ and seg QR.**

**Ans:** The union of ray PQ and ray QR is line QR.

**(iv) State the rays of which seg QR is a subset.**

**Ans:** Seg QR is a subset of ray QR, ray RQ, ray QS and ray QT.

**(v) Write the pair of opposite rays with common end point R.**

**Ans:** Ray RP and ray RS.

**(vi) Write any two rays with common endpoint S.**

**Ans:** Ray ST and ray SR.

**(vii) Write the intersection set of ray SP and ray ST.**

**Ans:** The intersection of ray SP and ray ST is point S.

**6. Answer the questions with the help of figure 1.14.**

**(i) State the points which are equidistant from point B. **

**Ans: **

(a) Point A and Point C.

(b) Point P and Point D.

**(ii) Write a pair of points equidistant from point Q.**

**Ans:** Point R and point P.

**(iii) Find d(U, V), d(P, C), d(V, B), d(U, L).**

**Solution:**

**d(U, V)**

The coordinate of point U is – 5.

The coordinate of point V is 5.

5 > – 5

d(U, V) = Greater coordinate – Smaller coordinate

d(U, V) = 5 – (– 5)

d(U, V) = 5 + 5

∴ d(U, V) = 10 Units

**d(P, C)**

The coordinate of point P is – 2.

The coordinate of point C is 4.

4 > – 2

d(P, C) = Greater coordinate – Smaller coordinate

d(P, C) = 4 – (– 2)

d(P, C) = 4 + 2

∴ d(P, C) = 6 Units

**d(V, B)**

The coordinate of point V is 5.

The coordinate of point B is 2.

5 > 2

d(V, B) = Greater coordinate – Smaller coordinate

∴ d(V, B) = 5 – 2

∴ d(V, B) = 3 Units

**d(U, L)**

The coordinate of point U is – 5.

The coordinate of point L is – 3.

– 3 > – 5

d(U, L) = Greater coordinate – Smaller coordinate

d(U, L) = – 3 – (– 5)

d(U, L) = – 3 + 5

∴ d(U, L) = 2 Units

**Ans: **

d(U, V) = 10 Units

d(P, C) = 6 Units

d(V, B) = 3 Units

d(U, L) = 2 Units

**Practice set 1.3**

**1. Write the following statements in ‘if–then’ form. **

**1. Write the following statements in ‘if–then’ form.**

**(i) The opposite angles of a parallelogram are congruent. **

**Ans:** If a quadrilateral is a parallelogram then its opposite angles are congruent.

** **

**(ii) The diagonals of a rectangle are congruent. **

**Ans:** If a quadrilateral is a rectangle then its diagonals are congruent.

** **

**(iii) In an isosceles triangle, the segment joining the vertex and the midpoint of the base is perpendicular to the base.**

**Ans:** If a triangle is an isosceles triangle, then the segment joining the vertex and midpoint of the base is perpendicular to the base.

**2. Write converses of the following statements. **

**2. Write converses of the following statements.**

**(i) The alternate angles formed by two parallel lines and their transversal are congruent.**

**Ans:** Converse statement:

**If alternate angles formed by two lines and its transversal are congruent then the lines are parallel.**

** **

**(ii) If a pair of the interior angles made by a transversal of two lines are supplementary then the lines are parallel.**

**Ans:** Converse statement:

**If two parallel lines are intersected by a transversal then interior angles so formed are supplementary.**

** **

**(iii) The diagonals of a rectangle are congruent.**

**Ans:** Converse statement:

**If the diagonals of a quadrilateral are congruent then that quadrilateral is rectangle.**

**Problem Set 1**

**Problem Set 1**

**1. Select the correct alternative from the answers of the questions given below.**

**1. Select the correct alternative from the answers of the questions given below.****(i) How many midpoints does a segment have?**

**(A) only one **

**(B) two **

**(C) three **

**(D) many**

** **

**Ans:** Option (A) : only one

** **

**(ii) How many points are there in the intersection of two distinct lines ?**

**(A) infinite **

**(B) two **

**(C) one **

**(D) not a single**

** **

**Ans:** Option (C) : one

** **

**(iii) How many lines are determined by three distinct points?**

**(A) two **

**(B) three **

**(C) one or three **

**(D) six**

** **

**Ans:** Option (C) : one or three

**(iv) Find d(A, B), if co-ordinates of A and B are – 2 and 5 respectively. **

**(A) – 2 **

**(B) 5 **

**(C) 7 **

**(D) 3**

** **

**Ans:** Option (C) : 7

** **

**Solution:**

**– 2 < 5**

**∴ d(A, B) = Greater co-ordinate – Smaller co-ordinate **

**∴ d(A, B) = 5 – (– 2)**

**∴ d(A, B) = 5 + 2**

**∴ d(A, B) = 7 units**

**(v) If P – Q – R and d(P, Q) = 2, d(P, R) = 10, then find d(Q, R).**

**(A) 12 **

**(B) 8 **

**(C) 96 **

**(D) 20**

** **

**Ans:** Option (B) : 8

** **

**Solution:**

**d(P, R) = d(P, Q) + d(Q, R) [∵ P – Q – R]**

**∴ 10 = 2 + d(Q, R)**

**∴ d(Q, R) = 10 – 2**

**∴ d(Q, R) = 8 units**

**2. On a number line, co-ordinates of P, Q, R are 3, – 5 and 6 respectively. State with reason whether the following statements are true or false.**

**2. On a number line, co-ordinates of P, Q, R are 3, – 5 and 6 respectively. State with reason whether the following statements are true or false.****Solution:**

**The co-ordinate of P is 3.**

**The co-ordinate of Q is – 5.**

**The co-ordinate of R is 6.**

** **

**3 > – 5**

** **

**∴ d(P, Q) = Greater co-ordinate – Smaller co-ordinate**

**∴ d(P, Q) = 3 – (– 5) **

**∴ d(P, Q) = 3 + 5 **

**∴ d(P, Q) = 8 units**

**– 5 < 6**

**∴ d(Q, R) = Greater co-ordinate – Smaller co-ordinate**

**∴ d(Q, R) = 6 – (– 5) **

**∴ d(Q, R) = 6 + 5 **

**∴ d(Q, R) = 11 units**

**6 > 3**

**∴ d(P, R) = Greater co-ordinate – Smaller co-ordinate**

**∴ d(P, R) = 6 – 3 **

**∴ d(P, R) = 3 units**

**(i) d(P, Q) + d(Q, R) = d(P, R) **

**Solution:**

**d(P, Q) + d(Q, R) = 8 + 11 **

**d(P, Q) + d(Q, R) = 19 units …(i)**

**d(P, R) = 3 units ….(ii)**

**∴ d(P, Q) + d(Q, R) ≠ d(P, R) … [From (i) and (ii)]**

** **

**∴ d(P, Q) + d(Q, R) = d(P, R) is a false statement.**

**(ii) d(P, R) + d(R, Q) = d(P, Q)**

**Solution:**

**d(P, R) + d(Q, R) = 3 + 11 **

**d(P, R) + d(Q, R) = 14 units …(i)**

**d(P, Q) = 8 units ….(ii)**

**∴ d(P, R) + d(R, Q) ≠ d(P, Q) … [From (i) and (ii)]**

**∴ d(P, R) + d(R, Q) = d(P, Q) is a false statement.**

**(iii) d(R, P) + d(P, Q) = d(R, Q) **

**Solution:**

**d(P, R) + d(P, Q) = 3 + 8 **

**d(P, R) + d(P, Q) = 11 units …(i)**

**d(R, Q) = 11 units ….(ii)**

**∴ d(P, R) + d(P, Q) = d(R, Q) … [From (i) and (ii)]**

**∴ d(P, R) + d(R, Q) = d(P, Q) is a true statement.**

**(iv) d(P, Q) – d(P, R) = d(Q, R)**

**Solution:**

**d(P, Q) – d(P, R) = 8 – 3 **

**d(P, Q) – d(P, R) = 5 units …(i)**

**d(R, Q) = 11 units ….(ii)**

**∴ d(P, R) + d(P, Q) ≠ d(R, Q) … [From (i) and (ii)]**

**∴ d(P, R) + d(R, Q) = d(P, Q) is a true statement.**

**3. Co-ordinates of some pairs of points are given below. Hence find the distance between each pair.**

**(i) 3, 6 **

**Solution:**

**Let,**

**The co-ordinate of point A be 3**

**The co-ordinate of point B be 6**

** **

**6 > 3**

** **

**d(A, B) = Greater co-ordinate – Smaller co-ordinate**

**∴ d(A, B) = 6 – 3**

**∴ d(A, B) = 3**

**∴ d(A, B) = 3**

** **

**Ans:** The distance between the given pair of points is 3 units.

** **

**(ii) – 9, – 1 **

**Solution:**

**Let,**

**The co-ordinate of point A be – 9**

**The co-ordinate of point B be – 1**

** **

**– 1 > – 9**

** **

**d(A, B) = Greater co-ordinate – Smaller co-ordinate**

**∴ d(A, B) = – 1 – (– 9)**

**∴ d(A, B) = – 1 + 9**

**∴ d(A, B) = 8**

**∴ d(A, B) = 8 units**

** **

**Ans:** The distance between the given pair of points is 8 units.

** **

**(iii) – 4, 5 **

**Solution:**

**Let,**

**The co-ordinate of point A be – 4**

**The co-ordinate of point B be 5**

** **

**5 > – 4**

** **

**d(A, B) = Greater co-ordinate – Smaller co-ordinate**

**∴ d(A, B) = 5 – (– 4)**

**∴ d(A, B) = 5 + 4**

**∴ d(A, B) = 9**

**∴ d(A, B) = 9 units**

** **

**Ans:** The distance between the given pair of points is 9 units.

** **

**(iv) 0, – 2**

**Solution:**

**Let,**

**The co-ordinate of point A be 0**

**The co-ordinate of point B be – 2**

** **

**0 > – 2**

** **

**d(A, B) = Greater co-ordinate – Smaller co-ordinate**

**∴ d(A, B) = 0 – (– 2)**

**∴ d(A, B) = 0 + 2**

**∴ d(A, B) = 2**

** **

**Ans:** The distance between the given pair of points is 2 units.

** **

**(v) x + 3, x – 3 **

**Solution:**

**Let,**

**The co-ordinate of point A be x + 3**

**The co-ordinate of point B be x – 3**

** **

**(x + 3) > (x – 3)**

** **

**d(A, B) = Greater co-ordinate – Smaller co-ordinate**

**∴ d(A, B) = (x + 3) – (x – 3)**

**∴ d(A, B) = x + 3 – x + 3**

**∴ d(A, B) = 6**

**∴ d(A, B) = 6 units**

** **

**Ans:** The distance between the given pair of points is 6 units.

** **

**(vi) – 25, – 47 **

**Solution:**

**Let,**

**The co-ordinate of A point be – 25 **

**The co-ordinate of B point be – 47**

** **

**– 25 > – 47 **

** **

**d(A, B) = Greater co-ordinate – Smaller co-ordinate**

**∴ d(A, B) = – 25 – (– 47)**

**∴ d(A, B) = – 25 + 47**

**∴ d(A, B) = 22 units**

** **

**Ans:** The distance between the given pair of points is 22 units.

** **

**(vii) 80, – 85**

**Solution:**

**Let,**

**The co-ordinate of point A be 80 **

**The co-ordinate of point B be – 85**

** **

**80 > – 85**

** **

**d(A, B) = Greater co-ordinate – Smaller co-ordinate**

**∴ d(A, B) = 80 – (– 85)**

**∴ d(A, B) = 80 + 85**

**∴ d(A, B) = 165 units**

** **

**Ans:** The distance between the given pair of points is 165 units.

**4. Co-ordinate of point P on a number line is – 7. Find the co-ordinates of points on the number line which are at a distance of 8 units from point P.**

**Solution:**

**Let Q = x be a point on the positive side of point P. **

**Co-ordinate of P is – 7**

** **

**x > – 7**

** **

** **

**d(P, Q) = x – (– 7)**

**∴ 8 = x + 7**

**∴ x = 8 – 7**

**∴ x = 1**

** **

**∴ Co-ordinate of point Q is 1.**

** **

**Let R = y be a point on the negative side of point P. **

**Co-ordinate of point P is – 7**

** **

**– 7 > y**

** **

**∴ d(P, R) = – 7 – y**

**∴ 8 = – 7 – y**

**∴ 8 + 7 = – y**

**∴ – y = 15**

**∴ y = – 15**

** **

**∴ Co-ordinate of point R is – 15**

** **

**Ans:** Co-ordinates of points at a distance of 8 units from point P is Q = 1 and R = – 15

**5. Answer the following questions.**

**5. Answer the following questions.****(i) If A – B – C and d(A, C) = 17, d(B, C) = 6.5 then d (A, B) = ?****Solution:**

d(A, C) = d(A, B) + d(B, C)* [∵ A – B – C]*

17 = d(A, B) + 6.5

d(A, B) = 17 – 6.5

d(A, B) = 10.5 units

**Ans:** d(A, B) is 10.5 units.

** **

**(ii) If P – Q – R and d(P, Q) = 3.4, d(Q, R)= 5.7 then d(P, R) = ?****Solution:**

d(P, R) = d(P, Q) + d(Q, R) *[∵ P – Q – R]*

d(P, R) = 3.4 + 5.7

d(P, R) = 9.1 units

**Ans:** d(P, R) is 9.1 units

**6. 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?**

**7. Write the following statements in conditional form.**

**7. Write the following statements in conditional form.****(i) Every rhombus is a square.**

**Ans:** If a quadrilateral is a square then it is a rhombus.

** **

**(ii) Angles in a linear pair are supplementary.**

**Ans:** If adjacent angles are supplementary, then they form a linear pair.

** **

**(iii) A triangle is a figure formed by three segments.**

**Ans:** If a polygon is three-sided closed figure, then it is a triangle.

** **

**(iv) A number having only two divisors is called a prime number.**

**Ans:** If a number is a prime number then it has only two divisors.

**8. Write the converse of each of the following statements.**

**8. Write the converse of each of the following statements.****(i) If the sum of measures of angles in a figure is 180⁰, then the figure is a triangle.**

**Ans: **If a figure is a triangle the sum of all angles of this figure is 180º.

** **

**(ii) If the sum of measures of two angles is 90⁰ then they are complement of each other.**

**Ans:** If two angles are complementary then their sum is 90º.

** **

**(iii) If the corresponding angles formed by a transversal of two lines are congruent then the two lines are parallel.**

**Ans: **If two parallel lines are intersected by a transversal then the pair of corresponding angles is congruent.

** **

**(iv) If the sum of the digits of a number is divisible by 3 then the number is divisible by 3.**

**Ans: **If a number is divisible by 3 then the sum of digits of this number is divisible by 3.

**9. Write the antecedent (given part) and the consequent (part to be proved) in the following statements.**

**9. Write the antecedent (given part) and the consequent (part to be proved) in the following statements.****(i) If all sides of a triangle are congruent then it’s all angles are congruent.**

**Ans:**

**Given: **

**In ∆ABC, **

**side AB ≅ side BC ≅ side AC**

** **

**To prove: **

**∠A ≅ ∠B ≅ ∠C**

** **

**(ii) The diagonals of a parallelogram bisect each other.**

**Ans:**

**Given: **

**□PQRS is a parallelogram**

**Diagonals PR and QS intersect at point M**

** **

**To prove:**

**(a) PM = RM **

**(b) QM = SM**

**10*. Draw a labelled figure showing information in each of the following statements and write the antecedent and the consequent.**

**(i) Two equilateral triangles are similar.**

**Ans: **

**Given: **

**In ∆ABC, **

**side AB ≅ side BC ≅ side AC**

**In ∆PQR, **

**side PQ ≅ side QR ≅ side PR**

** **

**To prove: **

**∆ABC ~ ∆PQR**

**(ii) If angles in a linear pair are congruent then each of them is a right angle.**

**Ans: **

**Given:**

**∠ABC and ∠ABD form a linear pair**

**∠ABC ≅ ∠ABD**

** **

**To prove: **

**∠ABC = ∠ABD = 90º**

**(iii) If the altitudes drawn on two sides of a triangle are congruent then those two sides are congruent.**

**Ans: **

**Given: **

**In ∆ABC,**

**seg BM ⊥ side AC, A – M – C**

**seg CN ⊥ side AB, A – N – B**

**seg BM ≅ seg CN**

** **

**To prove: **

**side AB ≅ side AC.**