Math  /  Geometry

QuestionPart 1 of 2 Points: 0 of 1 Save
Find the distance between the pair of points. Give an exact answer and a three-decimal-place approximation. (0,25) and (32,0)(0,-2 \sqrt{5}) \text { and }(3 \sqrt{2}, 0)

Studdy Solution

STEP 1

1. We are given two points in a 2-dimensional Cartesian coordinate system.
2. The coordinates of the points are (0,25)(0, -2\sqrt{5}) and (32,0)(3\sqrt{2}, 0).
3. We need to find both the exact distance and a three-decimal-place approximation.

STEP 2

1. Recall the distance formula for two points in a plane.
2. Substitute the given coordinates into the distance formula.
3. Simplify to find the exact distance.
4. Calculate the decimal approximation of the distance.

STEP 3

Recall the distance formula for two points (x1,y1)(x_1, y_1) and (x2,y2)(x_2, y_2):
d=(x2x1)2+(y2y1)2 d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}

STEP 4

Substitute the given coordinates (0,25)(0, -2\sqrt{5}) and (32,0)(3\sqrt{2}, 0) into the formula:
d=(320)2+(0(25))2 d = \sqrt{(3\sqrt{2} - 0)^2 + (0 - (-2\sqrt{5}))^2}

STEP 5

Simplify the expression to find the exact distance:
d=(32)2+(25)2 d = \sqrt{(3\sqrt{2})^2 + (2\sqrt{5})^2} =9×2+4×5 = \sqrt{9 \times 2 + 4 \times 5} =18+20 = \sqrt{18 + 20} =38 = \sqrt{38}

STEP 6

Calculate the decimal approximation of 38\sqrt{38}:
Using a calculator, 386.164\sqrt{38} \approx 6.164
The exact distance is 38\sqrt{38}, and the three-decimal-place approximation is 6.1646.164.

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