Chapter – 4 : Co-ordinate Geometry Distance Formula | Chapter Solution Class 9

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Ganit Prakash 2023 Co-ordinate Geometry Distance Formula Solution Class 9 Maths. Chapter 4 – Co-ordinate Geometry Distance Formula is provided here with simple step-by-step explanations. These solutions for Co-ordinate Geometry Distance Formula are extremely popular among class 9 students. The Co-ordinate Geometry Distance Formula solution is handy for quickly completing your homework and preparing for exams.

Book Name : Ganit Prakash
Subject : Mathematics (Maths)
Class : 9 (Madhyamik/WB)
Publisher : Prof. Nabanita Chatterjee
Chapter Name : Co-ordinate Geometry Distance Formula (4th Chapter)

 

Let us do ourselves – 4

Question 1

I measure the length of the straight line joining the following pairs of points –

(i) (18,0); (8,0)

(ii) (0,5); (0,4)

(iii) (-7,0); (-2,0)

(iv) (0,-10); (0,-3)

(v) (6,0); (-2,0)

(vi) (0,-5); (0,9)

(vii) (5,0); (0,10)

(viii) (3,0); (0,4)

(ix) (4,3); (2,1)

(x) (-2,-2); (2,2)

Solution:

Here, we use the formula to measure the length of the straight line

= \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}\\

(i) Distance = 18 – 8 =10 units (Ans)

(ii) Distance = 15 – 4 =11 (Ans)

(iii) Distance = -2-(-7) = -2 + 7 =5 units (Ans.)

(iv) Distance = -3 – (-10) = -3 + 10 = 7 units (Ans.)

(v) Distance = 6 – (-2) = 6 + 2 = 8 unit (Ans)

(vi) Distance = 9 -(-5) = 9 + 5 = 14 units (Ans.)

\text{ (vii) }\text { Distance } =\sqrt{(5-0)^2+(0-10)^2} \\

=\sqrt{25+100} \\

=\sqrt{125} \text {units} (Ans.)

\text { (viii) Distance }=\sqrt{(3-0)^2+(0-4)^2} \\

=\sqrt{(3)^2+(-4)^2} \\

=\sqrt{9+16} \\

=\sqrt{25}=5 \text { units} (Ans.)

\text {(ix) Distance } =\sqrt{(4-2)^2+(3-1)^2} \\

=\sqrt{(2)^2+(2)^2} \\

=\sqrt{4+4} \\

=\sqrt{8} \text { units} (Ans.)

\text{(x)\text Distance }=\sqrt{(-2-2)^2+(-2-2)^2} \\

=\sqrt{(-4)^2+(-4)^2} \\

=\sqrt{16+16} \\

=\sqrt{32} \text { units} (Ans.)

Let us work – 4

Question 1

Let us calculate the distances of the following points from the origin:

(i) (7, -24),

(ii) (3, -4),

(iii) (a + b, a – b)

Solution:

\text { Distance }=\sqrt{(7-0)^2+(-24-0)^2} \\

=\sqrt{(7)^2+(-24)^2} \\

=\sqrt{49+576} \\

=\sqrt{625}=25 \text { units} (Ans.)

\text { (ii) Distance }=\sqrt{(3-0)^2+(-4-0)^2} \\

=\sqrt{(3)^2+(-4)^2} \\

=\sqrt{9+16} \\

=\sqrt{25}=5 \text { units} (Ans.)

\text { (iii) }\text { Distance } =\sqrt{(\mathrm{a}+\mathrm{b}-0)^2+(\mathrm{a}-\mathrm{b}-0)^2} \\

=\sqrt{(\mathrm{a}+\mathrm{b})^2+(\mathrm{a}-\mathrm{b})^2} \\

=\sqrt{2\left(\mathrm{a}^2+\mathrm{b}^2\right)} \text { units} (Ans.)

Question 2

Let us calculate the distances between the two points given below:

(i) (5,7) and (8,3)

(ii) (7,0) and (2,-12)

(iii) (-\frac{3}{2}, 0) and (0,-2)

(iv) (3,6) and (-2,-6)

(v) (1,-3) and (8,3)

(vi) (5,7) and (8,3)

Solution:

\text {(i) Distance }=\sqrt{(5-8)^2+(7-3)^2} \\

=\sqrt{(-3)^2+(4)^2} \\

=\sqrt{9+16} \\

=\sqrt{25}=5 \text { units} (Ans.)

\text { (ii) Distance }=\sqrt{(7-2)^2+\{0-(-12)\}^2} \\

=\sqrt{(5)^2+(0+12)^2} \\

=\sqrt{25+144} \\

=\sqrt{169}=13 \text { units} (Ans.)

\text { (iii) Distance }=\sqrt{\left(-\frac{3}{2}-0\right)^2+\{0-(-2)\}^2} \\

=\sqrt{\left(-\frac{3}{2}\right)^2+(2)^2} \\

=\sqrt{\frac{9}{4}+4} \\

=\sqrt{\frac{9+16}{4}} \\

=\sqrt{\frac{25}{4}} \\

=\sqrt{5}\text{ units} (Ans.)

\text { (iv) Distance } =\sqrt{\{3-(-2)\}^2+\{6-(-6)\}^2} \\

=\sqrt{(3+2)^2+(6+6)^2} \\

=\sqrt{(5)^2+(12)^2} \\

=\sqrt{25+144}=13 \text {units} \\

=\sqrt{169}=13 \text { units} (Ans.)

\text { (v) Distance } =\sqrt{(1-8)^2+(-3-3)^2} \\

=\sqrt{(-7)^2+(-6)^2} \\

=\sqrt{49+36} \\

=\sqrt{85} \text { units (Ans.) } \\

\text{ (vi) Distance } =\sqrt{(5-8)^2+(7-3)^2} \\

=\sqrt{(-3)^2+(4)^2} \\

=\sqrt{9+16} \\

=\sqrt{25}=5 \text { units}(Ans.)

Question 3

Let us prove that point (-2, -11) is equidistant from the two points (-3, 7) and (4, 6)

Solution:

Let the coordinates of the points A, B & C be (-2, -11),(-3, 7) and (4, 6) respectively.

\mathrm{AB} = \sqrt{\left.\{(-2)-(-3)\}^2+\{(-11)-7)\right\}^2} \\

=\sqrt{(-2+3)^2+(-11-7)^2} \\

=\sqrt{(1)^2+(-18)^2} \\

=\sqrt{1+324}=\sqrt{325} \text { units (Ans.) } \\

\mathrm{AC} =\sqrt{\{4-(-2)\}^2+(-11-6)^2} \\

=\sqrt{(4+2)^2+(-17)^2} \\

=\sqrt{(6)^2+(-17)^2}\\

=\sqrt{36+289}=\sqrt{325} \text {(units Ans.)}\\

\therefore \quad \mathrm{AB} = \mathrm{AC}\\

Let us prove that point (-2, -11)

Hence, point A is equidistant from points B & C

Question 4

Let us show that the points (7,9),(3,-7) and (-3,3) are the vertices of a right-angled triangle by calculation.

Solution:

\text { The length of } \mathrm{AB}=\sqrt{(7-3)^{2}+\{9-(-7)\}^{2}}\\

=\sqrt{(4)^{2}+(9+7)^{2}}\\

=\sqrt{16+(16)^{2}}\\

=\sqrt{16+256}\\

=\sqrt{272} \text { units (Ans.) }\\

\therefore \quad \mathrm{AB}^{2}=272 \text { Units. }\\

Let us show that the points (7,9),(3,-7) and (-3,3)

\text { The length of } \mathrm{BC}=\sqrt{\{3-(-3)\}^{2}+(-7-3)^{2}}\\

=\sqrt{(3+3)^{2}+(-10)^{2}} \\

=\sqrt{(6)^{2}+(100)^{2}} \\

=\sqrt{36+100} \\

=\sqrt{136} \text { units (Ans.) } \\

\therefore \quad \mathrm{BC}^{2}=136 \text { Units }^{2}

\text { The length of } \mathrm{CA}=\sqrt{(-3-7)^{2}+(3-9)^{2}}\\

=\sqrt{(-10)^{2}+(-6)^{2}} \\

=\sqrt{100+36} \\

=\sqrt{136} \text { units (Ans.) }\\

\therefore \mathrm{CA}^{2}=136 \text { Units }^2\\

\therefore \quad \mathrm{AB}^{2}=\mathrm{BC}^{2}=\mathrm{CA}^{2}=136 \text { Units }^{2}\\

This shows that A, B & C i.e. the given points are the vertices of a right-angle triangle.

Question 5

Let us prove that in both of the following cases, the three points are the vertices of an isosceles triangle: (i) (1, 4), (4, 1) and (8, 8) (ii) (-2, -2), (2, 2) and (4, -4)

Solution:

Let A (1, 4), B (4, 1) and C (8, 8)

\mathrm{AB} =\sqrt{(1-4)^{2}+(4-1)^{2}} \\

=\sqrt{(-3)^{2}+(3)^{2}} \\

=\sqrt{9+9} \\

=\sqrt{18} \text { units }

Let us prove that in both of the following cases, the three points are the vertices of an isosceles triangle

\mathrm{BC} =\sqrt{(8-4)^{2}+(8-1)^{2}} \\

=\sqrt{(4)^{2}+(7)^{2}} \\

=\sqrt{16+49} \\

=\sqrt{65} \text { units } \\

\mathrm{CA} =\sqrt{(8-1)^{2}+(8-4)^{2}} \\

=\sqrt{(7)^{2}+(4)^{2}} \\

=\sqrt{49+16} \\

=\sqrt{65} \text { units } \\

\therefore \quad \mathrm{BC}=\mathrm{CA} ≠ \mathrm{AB} \\

This shows that A; B and C i.e. the given points are the vertices of an isosceles triangle.

(ii) Let A (-2, -2), B (2, 2) and C (4, -4)

\mathrm{AB} =\sqrt{(-2-2)^{2}+(-2-2)^{2}} \\

=\sqrt{(-4)^{2}+(-4)^{2}} \\

=\sqrt{16+16} \\

=\sqrt{32} \text { units } \\

Let us prove that in both of the following cases, the three points are the vertices of an isosceles triangle

\mathrm{BC} =\sqrt{(4-2)^{2}+(-4-2)^{2}} \\

=\sqrt{(2)^{2}+(-6)^{2}}\\

=\sqrt{4+36} \\

=\sqrt{40} \text { units } \\

\mathrm{CA} =\sqrt{\{4-(-2)\}^{2}+(-4-(-2)\}^{2}} \\

=\sqrt{(4+2)^{2}+(-4+2)^{2}} \\

=\sqrt{(6)^{2}+(-2)^{2}} \\

=\sqrt{36+4} \\

=\sqrt{40} \text { units } \\

\therefore \quad \mathrm{CA} = \mathrm{BC} ≠ \mathrm{AB}

This shows that A, B  and C i.e. the given points are the vertices of an isosceles triangle.

Question 6

Let us prove that the three points A(3,3), B(8,-2) and C(-2,-2) are the vertices of a right-angled triangle. Let us calculate the length of the hypotenuse of (\triangle ABC)

Solution:

\mathrm{AB}=\sqrt{(8-3)^{2}+(-2-3)^{2}} \\

=\sqrt{(5)^{2}+(-5)^{2}} \\

=\sqrt{25+25} \\

=\sqrt{50} \text { units } \\

\therefore \quad A B^{2}=50 \text { Units } \\

Let us prove that the three points A(3,3), B(8,-2) and C(-2,-2)

\mathrm{BC}=\sqrt{\{8-(-2)\}^{2}+\{-2-(-2)\}^{2}} \\

=\sqrt{(8+2)^{2}+(-2+2)^{2}} \\

=\sqrt{(10)^{2}+0} \\

=\sqrt{100 \text { units }} \\

\therefore \quad BC=100 \text { Units } \\

\mathrm{CA}=\sqrt{(-2-3)^{2}+(-2-3)^{2}} \\

=\sqrt{(-5)^{2}+(-5)^{2}}

=\sqrt{25+25} \\

=\sqrt{50} \text { units } \\

\therefore \quad \mathrm{CA}^{2}=50 \text { Units } \\

\therefore \quad \mathrm{BC}^{2}=\mathrm{AB}^{2}+\mathrm{CA}^{2}

This shows that the three points A (3, 3), B (8, -2) and C (-2, -2) are the vertices of a right-angled triangle.

Length of the hypotenuse of\triangle ABC = BC =\sqrt{100} \text{units}\\

= 10 (Ans)

Question 7

Let us show by calculation that the points (2, 1), (0, 0), (-1, 2) and (1, 3) are the angular points of a square.

Solution:

Let A (2, 1), B (0, 0), C (-1, 2) and D (1, 3)

\mathrm{AB} =\sqrt{(2-0)^{2}+(1-0)^{2}} \\

=\sqrt{(2)^{2}+(1)^{2}} \\

=\sqrt{4+1} \\

=\sqrt{5} \text { units } \\

Let us show by calculation that the points (2, 1), (0, 0), (-1, 2) and (1, 3)

\mathrm{BC} =\sqrt{(-1-0)^{2}+(2-0)^{2}} \\

=\sqrt{(-1)^{2}+(2)^{2}} \\

=\sqrt{1+4} \\

=\sqrt{5} \text { units }\\

\mathrm{CD} =\sqrt{(-1-1)^{2}+(2-3)^{2}} \\

=\sqrt{(-2)^{2}+(-1)^{2}} \\

=\sqrt{4+1} \\

=\sqrt{5} \text { units }\\

\mathrm{AD} =\sqrt{(2-1)^{2}+(1-3)^{2}} \\

=\sqrt{(1)^{2}+(-2)^{2}}\\

=\sqrt{1+4} \\

=\sqrt{5} \text { units } \\

\therefore \quad \mathrm{AB} =\mathrm{BC} =\mathrm {CD} = \mathrm{AD}

This shows that the four points A, B, C and D i.e. the given points are the vertices of a square.

Question 8

Let us calculate and see that for what value of 7, the distance of the two points (2, y) and (10, -2) will be 10.

Solution:

By the question,

\sqrt{(2-10)^{2}+(-9-y)^{2}}=10 \\

or, \sqrt{(-8)^{2}+(-9-y)^{2}}=10

Squaring both sides,

or, (- 8)2 + (- 9 - y)2 = 102

or, 64 + (- 9 - y)2 = 100

or, (- 9 - y)2 = 100 - 64

or, (- 9 - y)2 = 36

or, (- 9 - y)2 = 62

or, - 9 - y = ±6

Taking positive,

or, - 9 - y = 6

or, - y = - 6 + 9

or, - y = 15y = - 15

Taking negative

or, - 9 - y = 6

or, y = 6 + 9

or, y = 3

∴ y = - 3

∴ y = - 3, - 15 (Ans.)

Question 9

Let us find the point on the x-axis which is equidistant from the two points (3, 5) and (1, 3).

Solution:

Let the points on the x-axis be (x, 0) which is equidistant from the two points (3, 4) & (1, 3)

\therefore \sqrt{(x-3)^{2}+(5-0)^{2}}=\sqrt{(x-1)^{2}+(3-0)^{2}}\\

or, (x-3)^{2}+(5)^{2}=(x-1)^{2}+(3)^{2}

or, (x)^2-2.x.3 + (3)^2 + (5)^2 = (x)^2 - 2.x.1 + (1)^2 + (3)^2

or, x^2-6x +9 +25= x^2 - 2x + 1 + 9

or, -6x+25= -2x + 1

or, -6x+2x= 1-25

or, -4x=-24

x = {\frac{-24}{-4}= 6}

or, x = 6

∴ The required point is (6.0) (Ans.)

Question 10

Let us write by calculating whether the three points 0 (0, 0), A (4, 3) and B (8, 6) are collinear.

Solution:

\mathrm{OA} =\sqrt{(4-0)^{2}+(3-0)^{2}}\\

=\sqrt{4^{2}+3^{2}}\\

=\sqrt{16+9}\\

=\sqrt{25}=5\\

Let us write by calculation whether the three points 0 (0, 0), A (4, 3) and B (8, 6) are collinear

\mathrm{AB} =\sqrt{(8-4)^{2}+(6-3)^{2}}\\

=\sqrt{4^{2}+3^{2}}\\

=\sqrt{16+9}\\

=\sqrt{25}=5\\

\mathrm{OB} =\sqrt{(8-0)^{2}+(6-0)^{2}} \\

=\sqrt{8^{2}+6^{2}} \\

=\sqrt{64+36} \\

=\sqrt{100=10} \text { units }\\

\therefore \quad \mathrm{OA}+\mathrm{AB}=\mathrm{OB}

So, the points O (0, 0), A (4, 3) and B (8, 6) are collinear. (Proved)

Question 11

Let us show the three points (2, 2). (-2, -2) and (-2√3, 2√3) are the vertices of an equilateral triangle.

Solution:

\mathrm {AB} =\sqrt{(-2-2)^{2}+(-2-2)^{2}} \\

=\sqrt{(-4)^{2}+(-4)^{2}}\\

=\sqrt{16+16}\\

=\sqrt{32} \text { units }\\

\mathrm{BC} =\sqrt{(-2+2 \sqrt{3})^{2}+(-2-2 \sqrt{3})^{2}}

=\sqrt{(2 \sqrt{3}-2)^{2}+(2 \sqrt{3}+2)^{2}}\\

=\sqrt{2\left\{(2 \sqrt{ } 3)^{2}+(2)^{2}\right\}}\\

=\sqrt{2(12+4)}\\

=\sqrt{2 \times 16}\\

=\sqrt{32} \text { units } \\

\mathrm{CA} =\sqrt{(2+2 \sqrt{3})^{2}+(2-2 \sqrt{3})^{2}} \\

=\sqrt{2(2)^{2}+(2\sqrt{3})^{2}}

=\sqrt{2 .(4+12)}=\sqrt{2 \times 16}\\

=\sqrt{32 \text { units }} \\

\therefore \mathrm{AB} =\mathrm{BC}=\mathrm{CA}

This shows that the points A, B and C i.e. the given points are the vertices of an equilateral triangle.

Question 12

Let us show that the points (-7,2), (19,18), (15,-6) and(-11,-22) form a parallelogram when they are joined orderly.

Solution:

Let A (-7,2), B (19,18), C (15,-6) & D (-11,-22)

\mathrm{AB} =\sqrt{(19+7)^{2}+(18-2)^{2}}\\

=\sqrt{(26)^{2}+(16)^{2}} \\

=\sqrt{676+256}=\sqrt{932 \text { units }}\\

Let us show that the points (-7,2), (19,18), (15,-6) and(-11,-22)

\mathrm{BC} =\sqrt{(19-15)^{2}+(18+6)^{2}} \\

=\sqrt{(4)^{2}+(24)^{2}} \\

=\sqrt{16+476}=\sqrt{492 \text { units }} \\

\mathrm{CD} =\sqrt{(15+11)^{2}+(-6+22)^{2}} \\

=\sqrt{(26)^{2}+(6)^{2}} \\

=\sqrt{676+6}=\sqrt{932 \text { units }} \\

\mathrm{AD} =\sqrt{(-7+11)^{2}+(2+22)^{2}}

=\sqrt{(4)^{2}+(24)^{2}} \\

=\sqrt{16+426}=\sqrt{492} \text { units } \\

\therefore \quad \mathrm{AB} = \mathrm{CD} \text { and } \mathrm{BC}=\mathrm{AD}

This shows that the points A, B, C and D the given points are the vertices of a parallelogram (Proved).

Question 13

Let us show that the points (2,-2), (8,4), (5,7) and (-1,1) are the vertices of a rectangle.

Solution:

Let A (2,-2), B (8,4), C (5,7) & D (-1,1)

\mathrm{AB}=\sqrt{(2-8)^{2}+(-2-4)^{2}}\\

= \sqrt{(-6)^{2}+(-6)^{2}}\\

=\sqrt{36+36}=\sqrt{72 \text { units }}\\

\mathrm{BC}=\sqrt{(8-5)^{2}+(4-7)^{2}}\\

=\sqrt{(3)^{2}+(-3)^{2}} \\

=\sqrt{9+9}=\sqrt{18 \text { units }} \\

\mathrm{CD}=\sqrt{(-1-5)^{2}+(1-7)^{2}}\\

=\sqrt{(-6)^{2}+(-6)^{2}}\\

=\sqrt{36+36}=\sqrt{72} \text { units }\\

\mathrm{DA}=\sqrt{(-1-2)^{2}+(1+2)^{2}} \\

=\sqrt{(-3)^{2}+(3)^{2}} \\

=\sqrt{9+9}=\sqrt{18 \text { units }}\\

\therefore \quad \mathrm{AB}=\mathrm{CD} \text { and } \mathrm{BC}=\mathrm{DA}\\

This shows that the points A, B, C and D  i.e. the given points are the vertices of a rectangle (Proved).

Question 14

Let us show that the points (2,5), (5,9), (9,12) and (6,8) form a rhombus when they are joined orderly.

Solution:

Let A (2,5), B (5,9), C (9,12) & D (6,8)

\mathrm{AB} =\sqrt{(5-2)^{2}+(9-5)^{2}}\\

=\sqrt{(3)^{2}+(4)^{2}}\\

=\sqrt{9+16}=\sqrt{25}\\

=5 \text { units }\\

\mathrm{BC} =\sqrt{(9-5)^{2}+(12-9)^{2}} \\

=\sqrt{(4)^{2}+(3)^{2}} \\

=\sqrt{16+9}=\sqrt{25} \\

=5 \text { units } \\

\mathrm{CD} =\sqrt{(9-6)^{2}+(12-8)^{2}}\\

=\sqrt{(3)^{2}+(4)^{2}}\\

=\sqrt{9+16}=\sqrt{25}\\

=5 \text { units }\\

\mathrm{DA} =\sqrt{(6-2)^{2}+(8-5)^{2}} \\

=\sqrt{(4)^{2}+(3)^{2}} \\

=\sqrt{16+9}=\sqrt{25} \\

=5 \text { units }\\

\therefore \mathrm{AB}=\mathrm{BC}=\mathrm{CD}=\mathrm{DA}

Let us show that the points (2,5), (5,9), (9,12) and (6,8)

This shows that the points A, B, C  and D  i.e. the given points are the vertices of a Rhombus (Proved).

Multiple Choice Questions:

Question 15 (i)

The distance between the two points (a+b, c-d) and (a-b, c+d)  is

(a)2 \sqrt{a^{2}+c^{2}}

(b)2 \sqrt{b^{2}+d^{2}}

(c)\sqrt{a^{2}+c^{2}}

(d)\sqrt{b^{2}+d^{2}}

Solution:

(i)=\sqrt{\{(a+b)-(a-b)\}^{2}\{(c-d)-(c+d)\}^{2}} \\

=\sqrt{(a+b-a+b)^{2}+(c-d-c-d)^{2}} \\

=\sqrt{(2 a)^{2}+(-2 d)^{2}} \\

=\sqrt{4} \cdot \sqrt{b^{2}+d^{2}} \\

=2 \sqrt{b^{2}+d^{2}} \text { units. }

Question 15 (ii)

If the distance between the two points (x,-7) and (3,-3) is 5 units, then the values of x  are

(a) 0 or 6 (b) 2 or 3 (c) 5 or 1 (d) -6 or 0

Solution:

\quad \therefore Distance =5 units.\\

or, \sqrt{(x-3)^{2}+(-7+3)^{2}}=5\\

or, \sqrt{(x-3)^{2}+(-4)^{2}}=5[\mathrm{\because Squaring both sides}] \\

or, (x-3)^{2}+16=25\\

or, (x-3)^{2}=25-16=9\\

or, (x-3)^{2}=3^{2}\\

\therefore x-3= \pm 3\\

Taking positive,

x - 3 = 3

or, x = 3 + 3

or, x = 6

∴ x = 6

Taking negative

x - 3 = -3

or, x = -3 + 3

or, x = 0

∴ x = 0

∴ x = 0 or 6 (Ans.)

Question 15 (iii)

If the distance of the point (x, 4) from the origin is 5 units, then the values of x are

\mathrm{(a) \pm 4(b) \pm 5(c) \pm 3(d) none of these}

Solution:

\quad \therefore Distance =5 units.\\

or, \sqrt{(x-0)^{2}+(4-0)^{2}}=5\\

or, x^{2}+4^{2}=5 [\mathrm{\because Squaring both sides}]\\

or, x^{2}+16=25\\

or, x^{2}=25-16=9\\

or, \mathrm{x}^{2}=9 \quad \therefore x= \pm \sqrt{9}\\

\therefore x= \pm 3 (Ans.)\\

Question 15 (iv)

The triangle formed by the points (3,0), (-3,0) and (0,3) 

(a) equilateral (b) isosceles (c) scalene (d) isosceles right-angled

Solution:

\mathrm{Let} \mathrm{A}(3,0), \mathrm{B}(-3,0) and \mathrm{C}(0,3)\\

\mathrm{AB} =\sqrt{(-3-3)^{2}+(0-0)^{2}}\\

=\sqrt{(-6)^{2}+(0)^{2}} \\

=\sqrt{36}=6 \text { units } \\

The triangle formed by the points (3,0), (-3,0) and (0,3)

\mathrm{BC} =\sqrt{(-3-0)^{2}+(0-3)^{2}}\\

=\sqrt{(-3)^{2}+(-3)^{2}} \\

=\sqrt{9+9}=\sqrt{18 \text { units }} \\

\mathrm{AC} =\sqrt{(0-3)^{2}+(3-0)^{2}}=\sqrt{9+9} \\

=\sqrt{18} \text { units } \\

\therefore \mathrm{A}\mathrm{C}=\mathrm{B} \mathrm{C} ≠ \mathrm{AB}\\

This shows that A, B and C are the vertices of an isosceles triangle.

Question 15 (v)

The coordinates of the centre of the circle are (0, 0) and the coordinates of a point on the circumference are (3, 4), the length of the radius of the circle is

(a) 5 units (b) 4 units (c) 3 units (d) none of these

Solution:

= \sqrt{(3-0)^{2}+(4-0)^{2}} \\

=\sqrt{3^{2}+4^{2}} \\

=\sqrt{9+16}

=\sqrt{25}

=5 \text { units }

Short answer type questions:

Question 16 (i)

Let us write the value of y if the distance of the point (-4, y) from origin is 5 units.

Solution:

Distance = 5 units.

or, \sqrt{(-4-0)^{2}+(y-0)^{2}}=5\\

or, (-4)^{2}+y^{2}=5^{2}[\mathrm{\because Squaring both sides}] \\

or, 16+y^{2}=25\\

or, y^{2}=25-16\\

or, y^{2}=9

or, y= \pm \sqrt{ } 9 \quad \therefore y= \pm 3 (Ans).

Question 16 (ii)

Let us write the coordinates of a point on the y-axis which is equidistant from two points (2, 3) and (-1, 2).

Solution:

Let, \mathrm{A}(2,3) and \mathrm{B}(-1,2)\\

IMG 2030 removebg preview

Let a point on y-axis be C(0, y)\\

\therefore \mathrm{AC}=\mathrm{BC}\\

or, \mathrm{AC}^{2}=\mathrm{BC}^{2}\\

or, (2-0)^{2}+(3-y)^{2}=(-1-0)^{2}+(2-y)^{2}\\

or, (2)^{2}+(3)^{2}-2 \cdot 3 \cdot y+(y)^{2}=(-1)^{2}+(2)^{2}-2 \cdot 2 \cdot y+(y)^{2}\\

or, 4+9-6 y+y^{2}=1+4-4 y+y^{2}\\

or, 13-6 y=5-4 y\\

or, -6 y+4 y=5-13\\

or, -2 y=-8 \quad\\

or, y=\frac{-8}{-2}\\

\therefore \quad \mathrm{y}=4\\

The point (0, 4) on the y-axis is equidistant from two points (2, 3) and (-1, 2)

Question 16 (iii)

Let us write the coordinates of two points on x- the axis and y-axis for which an isosceles right-angled triangle is formed with the x-axis, y-axis and the straight line joining the two points.

Solution:

We can choose are y two points on the x-axis and y-anis as (2, 0) and (0, 2) & (3, 0) and (0,3) when these points are joined they formed an isosceles right-angled triangle with the co-ordinate.

Question 16 (iv)

Let us write the coordinates of two points on opposite sides of the x-axis which are equidistant from the x-axis.

Solution:

(i) (5,0) and (-5,0)

(ii) (10,0) and  (-10,0)

Question 16 (v)

Let us write the Co-ordinates of two points on opposite sides of the y-axis which are equidistant from the y-axis

Solution:

(i) (0, 5) and (0, -5)

(ii) (0, 10) and (0, -10)

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