Question

directions: find the distance between each pair of points.

1. (4, 5) and (3, -7)
2. (-6, -5) and (2, 0)
3. (-1, 4) and (1, -1)
4. (0, -8) and (3, -2)

5.
directions: find the coordinates of the mid - point of the segment given its endpoints.

1. a(5, 8) and b(-1, -4)
2. m(-5, 9) and n(-2, 7)
3. p(-3, -7) and q(3, -5)
4. f(2, -6) and g(-8, 5)

Explanation:

1. For the distance - formula between two points \((x_1,y_1)\) and \((x_2,y_2)\) is \(d = \sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}\)

• Question 1: Points \((4,5)\) and \((3, - 7)\)
• # Explanation:

Step1: Identify \(x_1,y_1,x_2,y_2\)

\(x_1 = 4,y_1 = 5,x_2 = 3,y_2=-7\)

Step2: Substitute into distance formula

\(d=\sqrt{(3 - 4)^2+(-7 - 5)^2}=\sqrt{(-1)^2+(-12)^2}=\sqrt{1 + 144}=\sqrt{145}\)

• # Answer:

\(\sqrt{145}\)

• Question 2: Points \((-6,-5)\) and \((2,0)\)
• # Explanation:

Step1: Identify \(x_1,y_1,x_2,y_2\)

\(x_1=-6,y_1 = - 5,x_2 = 2,y_2 = 0\)

Step2: Substitute into distance formula

\(d=\sqrt{(2+6)^2+(0 + 5)^2}=\sqrt{8^2+5^2}=\sqrt{64 + 25}=\sqrt{89}\)

• # Answer:

\(\sqrt{89}\)

• Question 3: Points \((-1,4)\) and \((1,-1)\)
• # Explanation:

Step1: Identify \(x_1,y_1,x_2,y_2\)

\(x_1=-1,y_1 = 4,x_2 = 1,y_2=-1\)

Step2: Substitute into distance formula

\(d=\sqrt{(1 + 1)^2+(-1 - 4)^2}=\sqrt{2^2+(-5)^2}=\sqrt{4 + 25}=\sqrt{29}\)

• # Answer:

\(\sqrt{29}\)

• Question 4: Points \((0,-8)\) and \((3,-2)\)
• # Explanation:

Step1: Identify \(x_1,y_1,x_2,y_2\)

\(x_1 = 0,y_1=-8,x_2 = 3,y_2=-2\)

Step2: Substitute into distance formula

\(d=\sqrt{(3 - 0)^2+(-2 + 8)^2}=\sqrt{3^2+6^2}=\sqrt{9+36}=\sqrt{45}=3\sqrt{5}\)

• # Answer:

\(3\sqrt{5}\)

1. For the mid - point formula between two points \((x_1,y_1)\) and \((x_2,y_2)\) is \(M=(\frac{x_1 + x_2}{2},\frac{y_1 + y_2}{2})\)

• Question 6: Points \(A(5,8)\) and \(B(-1,-4)\)
• # Explanation:

Step1: Identify \(x_1,y_1,x_2,y_2\)

\(x_1 = 5,y_1 = 8,x_2=-1,y_2=-4\)

Step2: Substitute into mid - point formula

\(M=(\frac{5-1}{2},\frac{8 - 4}{2})=(2,2)\)

• # Answer:

\((2,2)\)

• Question 7: Points \(M(-5,9)\) and \(N(-2,7)\)
• # Explanation:

Step1: Identify \(x_1,y_1,x_2,y_2\)

\(x_1=-5,y_1 = 9,x_2=-2,y_2 = 7\)

Step2: Substitute into mid - point formula

\(M=(\frac{-5-2}{2},\frac{9 + 7}{2})=(-\frac{7}{2},8)\)

• # Answer:

\((-\frac{7}{2},8)\)

• Question 8: Points \(P(-3,-7)\) and \(Q(3,-5)\)
• # Explanation:

Step1: Identify \(x_1,y_1,x_2,y_2\)

\(x_1=-3,y_1=-7,x_2 = 3,y_2=-5\)

Step2: Substitute into mid - point formula

\(M=(\frac{-3 + 3}{2},\frac{-7-5}{2})=(0,-6)\)

• # Answer:

\((0,-6)\)

• Question 9: Points \(F(2,-6)\) and \(G(-8,5)\)
• # Explanation:

Step1: Identify \(x_1,y_1,x_2,y_2\)

\(x_1 = 2,y_1=-6,x_2=-8,y_2 = 5\)

Step2: Substitute into mid - point formula

\(M=(\frac{2-8}{2},\frac{-6 + 5}{2})=(-3,-\frac{1}{2})\)

• # Answer:

\((-3,-\frac{1}{2})\)

Answer:

1. For the distance - formula between two points \((x_1,y_1)\) and \((x_2,y_2)\) is \(d = \sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}\)

• Question 1: Points \((4,5)\) and \((3, - 7)\)
• # Explanation:

Step1: Identify \(x_1,y_1,x_2,y_2\)

\(x_1 = 4,y_1 = 5,x_2 = 3,y_2=-7\)

Step2: Substitute into distance formula

\(d=\sqrt{(3 - 4)^2+(-7 - 5)^2}=\sqrt{(-1)^2+(-12)^2}=\sqrt{1 + 144}=\sqrt{145}\)

• # Answer:

\(\sqrt{145}\)

• Question 2: Points \((-6,-5)\) and \((2,0)\)
• # Explanation:

Step1: Identify \(x_1,y_1,x_2,y_2\)

\(x_1=-6,y_1 = - 5,x_2 = 2,y_2 = 0\)

Step2: Substitute into distance formula

\(d=\sqrt{(2+6)^2+(0 + 5)^2}=\sqrt{8^2+5^2}=\sqrt{64 + 25}=\sqrt{89}\)

• # Answer:

\(\sqrt{89}\)

• Question 3: Points \((-1,4)\) and \((1,-1)\)
• # Explanation:

Step1: Identify \(x_1,y_1,x_2,y_2\)

\(x_1=-1,y_1 = 4,x_2 = 1,y_2=-1\)

Step2: Substitute into distance formula

\(d=\sqrt{(1 + 1)^2+(-1 - 4)^2}=\sqrt{2^2+(-5)^2}=\sqrt{4 + 25}=\sqrt{29}\)

• # Answer:

\(\sqrt{29}\)

• Question 4: Points \((0,-8)\) and \((3,-2)\)
• # Explanation:

Step1: Identify \(x_1,y_1,x_2,y_2\)

\(x_1 = 0,y_1=-8,x_2 = 3,y_2=-2\)

Step2: Substitute into distance formula

\(d=\sqrt{(3 - 0)^2+(-2 + 8)^2}=\sqrt{3^2+6^2}=\sqrt{9+36}=\sqrt{45}=3\sqrt{5}\)

• # Answer:

\(3\sqrt{5}\)

1. For the mid - point formula between two points \((x_1,y_1)\) and \((x_2,y_2)\) is \(M=(\frac{x_1 + x_2}{2},\frac{y_1 + y_2}{2})\)

• Question 6: Points \(A(5,8)\) and \(B(-1,-4)\)
• # Explanation:

Step1: Identify \(x_1,y_1,x_2,y_2\)

\(x_1 = 5,y_1 = 8,x_2=-1,y_2=-4\)

Step2: Substitute into mid - point formula

\(M=(\frac{5-1}{2},\frac{8 - 4}{2})=(2,2)\)

• # Answer:

\((2,2)\)

• Question 7: Points \(M(-5,9)\) and \(N(-2,7)\)
• # Explanation:

Step1: Identify \(x_1,y_1,x_2,y_2\)

\(x_1=-5,y_1 = 9,x_2=-2,y_2 = 7\)

Step2: Substitute into mid - point formula

\(M=(\frac{-5-2}{2},\frac{9 + 7}{2})=(-\frac{7}{2},8)\)

• # Answer:

\((-\frac{7}{2},8)\)

• Question 8: Points \(P(-3,-7)\) and \(Q(3,-5)\)
• # Explanation:

Step1: Identify \(x_1,y_1,x_2,y_2\)

\(x_1=-3,y_1=-7,x_2 = 3,y_2=-5\)

Step2: Substitute into mid - point formula

\(M=(\frac{-3 + 3}{2},\frac{-7-5}{2})=(0,-6)\)

• # Answer:

\((0,-6)\)

• Question 9: Points \(F(2,-6)\) and \(G(-8,5)\)
• # Explanation:

Step1: Identify \(x_1,y_1,x_2,y_2\)

\(x_1 = 2,y_1=-6,x_2=-8,y_2 = 5\)

Step2: Substitute into mid - point formula

\(M=(\frac{2-8}{2},\frac{-6 + 5}{2})=(-3,-\frac{1}{2})\)

• # Answer:

\((-3,-\frac{1}{2})\)