Given R(-6,19) and S(14,-6), what are the coordinates of the point on segment RS two-fifths of the distance from R and S?
let that point be P(x,y)
the way I read your question is ...
the ratio of RP : PS = 2 : 3
for the x-coordinate:
(x+6)/(14+6) = 2/5
x+6 = 20(2/5) = 8
x = 2
do the y in the same way.
Ah, let me do some math here. If we have a segment RS, and we want to find a point on this segment that is two-fifths of the distance from R to S... *calculates* Okay, got it!
So, we start at point R(-6, 19) and move two-fifths of the way towards S(14, -6). Since two-fifths of a distance is the same as 2 out of every 5 parts, we can imagine dividing the distance between R and S into five equal parts and then moving two parts towards S. Let's do that!
First, we find the difference in x-coordinates and y-coordinates of R and S. The difference in x-coordinates is 14 - (-6) = 20, and the difference in y-coordinates is -6 - 19 = -25.
Now, we'll take two-fifths of these differences to determine how much we should move towards S. For the x-coordinate, we have 2/5 * 20 = 8, and for the y-coordinate, we have 2/5 * (-25) = -10.
Finally, we add these calculated values to the x-coordinate and y-coordinate of R to find the coordinates of the point two-fifths of the distance from R to S.
So, the coordinates of the desired point are -6 + 8 = 2 for x, and 19 - 10 = 9 for y. Therefore, the point on segment RS, two-fifths of the distance from R and S, is (2, 9).
Hope that helps! And remember, even math problems can be solved with a little bit of clowning around! 🤡
To find the coordinates of the point on segment RS that is two-fifths of the distance from R and S, we need to first find the distance between points R and S.
Let's use the distance formula, which is given by:
d = √((x2 - x1)^2 + (y2 - y1)^2)
Substituting the coordinates of R(-6,19) and S(14,-6) into the formula, we have:
d = √((14 - (-6))^2 + (-6 - 19)^2)
= √((20)^2 + (-25)^2)
= √(400 + 625)
= √(1025)
≈ 32.02
Now, we can find the two-fifths distance from R to S. Multiplying the total distance by the fraction 2/5, we get:
distance = (2/5) * 32.02
≈ 12.81
To find the coordinates of the point that is two-fifths of the distance from R and S, we need to move 12.81 units from R towards S. We'll move this distance in the x-direction and y-direction.
The x-coordinate of the new point = -6 + (12.81 * (14 - (-6)) / 32.02) ≈ -1.35
The y-coordinate of the new point = 19 + (12.81 * (-6 - 19) / 32.02) ≈ 4.66
Therefore, the coordinates of the point on segment RS that is two-fifths of the distance from R and S are approximately (-1.35, 4.66).
To find the coordinates of a point that is two-fifths of the distance from R to S, we need to use the concept of the section formula.
The section formula states that if we have two points with coordinates (x1, y1) and (x2, y2) and we want to find a point that divides the line segment joining these two points in a ratio of m:n, then the coordinates of the point can be found using the formula:
Px = (mx2 + nx1) / (m + n)
Py = (my2 + ny1) / (m + n)
In this case, R has coordinates (-6, 19) and S has coordinates (14, -6). We want to find the coordinates of a point that is two-fifths of the distance from R to S. Therefore, m = 2 and n = 5.
Using the section formula, we can calculate the coordinates of the desired point:
Px = (2*14 + 5*(-6)) / (2 + 5)
Py = (2*(-6) + 5*19) / (2 + 5)
Simplifying the expressions, we get:
Px = (28 - 30) / 7 = -2 / 7
Py = (-12 + 95) / 7 = 83 / 7
Therefore, the coordinates of the point on segment RS that is two-fifths of the distance from R to S are (-2/7, 83/7).