Find the distance between the following points .
(i) A(0,0) ,B (-5,12)
(ii) M (-4,-3) , N (2,-1)
(iii) P (3,-4) ,Q(-3,4)
(iv) L(4,1) , M (1,-3)
(v) P(-1,1) ,Q(5,-7)

Answer

To find the distance between the given points, we will use the distance formula:

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

where \( (x_1, y_1) \) and \( (x_2, y_2) \) are the coordinates of the two points.

### (i) A(0,0), B(-5,12)

1. Identify the coordinates:
- \( A(0, 0) \) → \( (x_1, y_1) = (0, 0) \)
- \( B(-5, 12) \) → \( (x_2, y_2) = (-5, 12) \)

2. Apply the distance formula:
\[ d = \sqrt{((-5) - 0)^2 + (12 - 0)^2} \]
\[ = \sqrt{(-5)^2 + (12)^2} \]
\[ = \sqrt{25 + 144} \]
\[ = \sqrt{169} \]
\[ = 13 \]

**Distance between A and B is 13 units.**

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### (ii) M(-4,-3), N(2,-1)

1. Identify the coordinates:
- \( M(-4, -3) \) → \( (x_1, y_1) = (-4, -3) \)
- \( N(2, -1) \) → \( (x_2, y_2) = (2, -1) \)

2. Apply the distance formula:
\[ d = \sqrt{(2 - (-4))^2 + (-1 - (-3))^2} \]
\[ = \sqrt{(2 + 4)^2 + (-1 + 3)^2} \]
\[ = \sqrt{6^2 + 2^2} \]
\[ = \sqrt{36 + 4} \]
\[ = \sqrt{40} \]
\[ = \sqrt{4 \times 10} = 2\sqrt{10} \]

**Distance between M and N is \( 2\sqrt{10} \) units.**

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### (iii) P(3,-4), Q(-3,4)

1. Identify the coordinates:
- \( P(3, -4) \) → \( (x_1, y_1) = (3, -4) \)
- \( Q(-3, 4) \) → \( (x_2, y_2) = (-3, 4) \)

2. Apply the distance formula:
\[ d = \sqrt{((-3) - 3)^2 + (4 - (-4))^2} \]
\[ = \sqrt{(-6)^2 + (4 + 4)^2} \]
\[ = \sqrt{36 + 64} \]
\[ = \sqrt{100} \]
\[ = 10 \]

**Distance between P and Q is 10 units.**

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### (iv) L(4,1), M(1,-3)

1. Identify the coordinates:
- \( L(4, 1) \) → \( (x_1, y_1) = (4, 1) \)
- \( M(1, -3) \) → \( (x_2, y_2) = (1, -3) \)

2. Apply the distance formula:
\[ d = \sqrt{(1 - 4)^2 + (-3 - 1)^2} \]
\[ = \sqrt{(-3)^2 + (-4)^2} \]
\[ = \sqrt{9 + 16} \]
\[ = \sqrt{25} \]
\[ = 5 \]

**Distance between L and M is 5 units.**

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### (v) P(-1,1), Q(5,-7)

1. Identify the coordinates:
- \( P(-1, 1) \) → \( (x_1, y_1) = (-1, 1) \)
- \( Q(5, -7) \) → \( (x_2, y_2) = (5, -7) \)

2. Apply the distance formula:
\[ d = \sqrt{(5 - (-1))^2 + (-7 - 1)^2} \]
\[ = \sqrt{(5 + 1)^2 + (-8)^2} \]
\[ = \sqrt{6^2 + 64} \]
\[ = \sqrt{36 + 64} \]
\[ = \sqrt{100} \]
\[ = 10 \]

**Distance between P and Q is 10 units.**

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