Math  /  Geometry

QuestionFind the coordinates of PP that represent the weighted average for the given set of points with the given weights. - A has a weight of 3 . - BB has a weight of 2 .

Studdy Solution

STEP 1

1. We are given two points A A and B B with their coordinates and respective weights.
2. The coordinates of A A are (1,1) (-1, 1) and the weight of A A is 3.
3. The coordinates of B B are (3,9) (3, 9) and the weight of B B is 2.
4. The weighted average of the points can be calculated using the formula for weighted averages of coordinates.

STEP 2

1. Calculate the weighted average of the x-coordinates of points A A and B B .
2. Calculate the weighted average of the y-coordinates of points A A and B B .
3. Combine the weighted averages of the x and y coordinates to find the coordinates of point P P .

STEP 3

Calculate the total weight.
Total weight=3+2=5 \text{Total weight} = 3 + 2 = 5

STEP 4

Calculate the weighted sum of the x-coordinates.
Weighted sum of x-coordinates=(31)+(23)=3+6=3 \text{Weighted sum of x-coordinates} = (3 \cdot -1) + (2 \cdot 3) = -3 + 6 = 3

STEP 5

Calculate the weighted average of the x-coordinates.
Weighted average of x-coordinates=Weighted sum of x-coordinatesTotal weight=35=0.6 \text{Weighted average of x-coordinates} = \frac{\text{Weighted sum of x-coordinates}}{\text{Total weight}} = \frac{3}{5} = 0.6

STEP 6

Calculate the weighted sum of the y-coordinates.
Weighted sum of y-coordinates=(31)+(29)=3+18=21 \text{Weighted sum of y-coordinates} = (3 \cdot 1) + (2 \cdot 9) = 3 + 18 = 21

STEP 7

Calculate the weighted average of the y-coordinates.
Weighted average of y-coordinates=Weighted sum of y-coordinatesTotal weight=215=4.2 \text{Weighted average of y-coordinates} = \frac{\text{Weighted sum of y-coordinates}}{\text{Total weight}} = \frac{21}{5} = 4.2

STEP 8

Combine the weighted averages of the x and y coordinates to find the coordinates of point P P .
P=(0.6,4.2) P = (0.6, 4.2)
Solution: The coordinates of point P P are (0.6,4.2) (0.6, 4.2) .

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