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QuestionFind the coordinates of a station such that the distance from Rolesville R(1,2)R(1,2) to the station and from the station to Townsend T(8,23)T(8,23) is in a 3:43:4 ratio.

Studdy Solution

STEP 1

Assumptions1. The coordinates of Rolesville (R) are (1,) . The coordinates of Townsend () are (8,23)
3. The ratio of the distance from Rolesville to the station (RS) and the distance from the station to Townsend () is34

STEP 2

First, we need to find the total distance between Rolesville and Townsend. We can do this by using the distance formula in two dimensions, which isd=(x2−x1)2+(y2−y1)2d = \sqrt{(x2 - x1)^2 + (y2 - y1)^2}

STEP 3

Now, plug in the given values for the coordinates of Rolesville and Townsend to calculate the distance.
d=(8−1)2+(23−2)2d = \sqrt{(8 -1)^2 + (23 -2)^2}

STEP 4

Calculate the distance.
d=(7)2+(21)2=49+441=490d = \sqrt{(7)^2 + (21)^2} = \sqrt{49 +441} = \sqrt{490}

STEP 5

Now that we have the total distance, we can find the distance from Rolesville to the station (RS) and the distance from the station to Townsend () using the given ratio34. The sum of the ratio34 is7, so the distance RS is3/7 of the total distance, and the distance is4/7 of the total distance.
RS=37dRS = \frac{3}{7}d=47d = \frac{4}{7}d

STEP 6

Plug in the value for the total distance to calculate the distances RS and.
RS=3490RS = \frac{3}{}\sqrt{490}=4490 = \frac{4}{}\sqrt{490}

STEP 7

Now, we need to find the coordinates of the station. We can do this by using the formula for dividing a line segment in a given ratio. The formula is(x,y)=(mx2+nx1m+n,my2+ny1m+n)(x, y) = \left(\frac{mx2 + nx1}{m + n}, \frac{my2 + ny1}{m + n}\right)where m and n are the given ratio, and (x1, y1) and (x2, y2) are the coordinates of the two points.

STEP 8

Now, plug in the given values for the coordinates of Rolesville and Townsend and the ratio to calculate the coordinates of the station.
(x,y)=(3⋅8+4⋅13+4,3⋅23+4⋅23+4)(x, y) = \left(\frac{3 \cdot8 +4 \cdot1}{3 +4}, \frac{3 \cdot23 +4 \cdot2}{3 +4}\right)

STEP 9

Calculate the coordinates of the station.
(x,y)=(24+47,69+87)=(4,11)(x, y) = \left(\frac{24 +4}{7}, \frac{69 +8}{7}\right) = (4,11)The coordinates of the new station should be (4,11).

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