Math  /  Geometry

QuestionIn Exercises 32-33, find the exact distance between the two points.
32. G(1,5)G(-1,-5) and H(3,8)GH=H(3,-8) \quad G H= \qquad
33. G(2,5)G(2,5) and H(4,1)GH=H(4,-1) G H= \qquad

Use the information to find the midpoint between points AA and BB. In a three-dimensional coordinate system, the midpoint between A(x1,y1,z1)A\left(x_{1}, y_{1}, z_{1}\right) and B(x2,y2,z2)B\left(x_{2}, y_{2}, z_{2}\right) is (x1+x22,y1+y22,z1+z22)\left(\frac{x_{1}+x_{2}}{2}, \frac{y_{1}+y_{2}}{2}, \frac{z_{1}+z_{2}}{2}\right).
34. A(3,0,4)A(-3,0,4) B(7,2,8)B(7,2,8)

Use the information to find the distance between points AA and BB. Leave answers in simplified radical form. In a threedimensional coordinate system, the distance between the two points A(x1,y1,z1)A\left(x_{1}, y_{1}, z_{1}\right) and B(x2,y2,z2)B\left(x_{2}, y_{2}, z_{2}\right) is AB=(x2x1)2+(y2y1)2+(z2z1)2A B=\sqrt{\left(x_{2}-x_{1}\right)^{2}+\left(y_{2}-y_{1}\right)^{2}+\left(z_{2}-z_{1}\right)^{2}}
35. A(2,1,2)A(-2,1,2) B(2,3,4)B(2,-3,4)

Studdy Solution

STEP 1

1. The distance between two points in a 2-dimensional coordinate system can be found using the distance formula: d=(x2x1)2+(y2y1)2 d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}
2. The midpoint between two points in a 3-dimensional coordinate system can be found using the midpoint formula: M=(x1+x22,y1+y22,z1+z22) M = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}, \frac{z_1 + z_2}{2} \right)
3. The distance between two points in a 3-dimensional coordinate system can be found using the distance formula: d=(x2x1)2+(y2y1)2+(z2z1)2 d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2 + (z_2 - z_1)^2}

STEP 2

1. Calculate the distance between points G(1,5)G(-1,-5) and H(3,8)H(3,-8) in 2D.
2. Calculate the distance between points G(2,5)G(2,5) and H(4,1)H(4,-1) in 2D.
3. Calculate the midpoint between points A(3,0,4)A(-3,0,4) and B(7,2,8)B(7,2,8) in 3D.
4. Calculate the distance between points A(2,1,2)A(-2,1,2) and B(2,3,4)B(2,-3,4) in 3D.

STEP 3

Use the distance formula to find the distance between G(1,5)G(-1,-5) and H(3,8)H(3,-8). d=(3(1))2+(8(5))2 d = \sqrt{(3 - (-1))^2 + (-8 - (-5))^2}

STEP 4

Simplify the expressions inside the square root. d=(3+1)2+(8+5)2 d = \sqrt{(3 + 1)^2 + (-8 + 5)^2}

STEP 5

Calculate the squares of the simplified terms. d=42+(3)2 d = \sqrt{4^2 + (-3)^2}

STEP 6

Sum the squares and take the square root. d=16+9 d = \sqrt{16 + 9} d=25 d = \sqrt{25} d=5 d = 5

STEP 7

Use the distance formula to find the distance between G(2,5)G(2,5) and H(4,1)H(4,-1). d=(42)2+(15)2 d = \sqrt{(4 - 2)^2 + (-1 - 5)^2}

STEP 8

Simplify the expressions inside the square root. d=(42)2+(15)2 d = \sqrt{(4 - 2)^2 + (-1 - 5)^2} d=22+(6)2 d = \sqrt{2^2 + (-6)^2}

STEP 9

Calculate the squares of the simplified terms. d=22+(6)2 d = \sqrt{2^2 + (-6)^2} d=4+36 d = \sqrt{4 + 36}

STEP 10

Sum the squares and take the square root. d=40 d = \sqrt{40} d=210 d = 2\sqrt{10}

STEP 11

Use the midpoint formula to find the midpoint between A(3,0,4)A(-3,0,4) and B(7,2,8)B(7,2,8). M=(3+72,0+22,4+82) M = \left( \frac{-3 + 7}{2}, \frac{0 + 2}{2}, \frac{4 + 8}{2} \right)

STEP 12

Simplify the expressions inside the parentheses. M=(42,22,122) M = \left( \frac{4}{2}, \frac{2}{2}, \frac{12}{2} \right)

STEP 13

Calculate the midpoint coordinates. M=(2,1,6) M = (2, 1, 6)

STEP 14

Use the distance formula to find the distance between A(2,1,2)A(-2,1,2) and B(2,3,4)B(2,-3,4). d=(2(2))2+(31)2+(42)2 d = \sqrt{(2 - (-2))^2 + (-3 - 1)^2 + (4 - 2)^2}

STEP 15

Simplify the expressions inside the square root. d=(2+2)2+(31)2+(42)2 d = \sqrt{(2 + 2)^2 + (-3 - 1)^2 + (4 - 2)^2} d=42+(4)2+22 d = \sqrt{4^2 + (-4)^2 + 2^2}

STEP 16

Calculate the squares of the simplified terms. d=16+16+4 d = \sqrt{16 + 16 + 4}

STEP 17

Sum the squares and take the square root. d=36 d = \sqrt{36} d=6 d = 6
Solution:
1. The distance between G(1,5)G(-1,-5) and H(3,8)H(3,-8) is 55.
2. The distance between G(2,5)G(2,5) and H(4,1)H(4,-1) is 2102\sqrt{10}.
3. The midpoint between A(3,0,4)A(-3,0,4) and B(7,2,8)B(7,2,8) is (2,1,6)(2, 1, 6).
4. The distance between A(2,1,2)A(-2,1,2) and B(2,3,4)B(2,-3,4) is 66.

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