Answer:
The coordinates of point R are (x, y) = (9/10, 8).
Explanation:
To find the coordinates of point R, we need to set up the ratios of the distances between the points using the given information.
We know that 5 MR : RP, and we know the coordinates of points M
and P.
Let's denote the distance between M and R as MR = d1 and the distance between R and P as RP = d2.
We can then set up the following equation:
5 d1 : d2 = (-10) - (-2) : 11 - 8
Simplifying the equation, we get:
5 d1/d2 = 4/3
Cross-multiplying to solve for d2, we get:
3d2 = 20d1
Dividing both sides by 3, we get:
d2 = (20/3) d1
Now, we can use the distance formula to find the distances d1 and d2 in terms of the coordinates of points M, R, and P.
The distance formula is:
d = √((x2 - x1)^2 + (y2 - y1)^2)
Using the formula to find d1, we get:
d1 = √((x - (-10))^2 + (y - 8)^2)
d1 = √((x + 10)^2 + (y - 8)^2)
Similarly, using the formula to find d2, we get:
d2 = √((x - (-2))^2 + (y - 11)^2)
d2 = √((x + 2)^2 + (y - 11)^2)
Substituting d2 = (20/3) d1 into the second equation, we get:
√((x + 2)^2 + (y - 11)^2) = (20/3) √((x + 10)^2 + (y - 8)^2)
Squaring both sides of the equation, we get:
(x + 2)^2 + (y - 11)^2 = (400/9) ((x + 10)^2 + (y - 8)^2)
Expanding both sides, we get:
x^2 + 4x + 4 + y^2 - 22y + 121 = (400/9) (x^2 + 20x + 100 + y^2 - 16y + 64)
Distributing the (400/9) on the right side, we get:
x^2 + 4x + 4 + y^2 - 22y + 121 = (400/9) x^2 + (8000/9) x + (6400/9) + (400/9) y^2 - (3200/9) y + (5760/9)
Combining like terms on both sides, we get:
0 = (400/9) x^2 - (364/9) x + (1600/9) y^2 - (1722/9) y + (1776/9)
To solve for x and y, we can use a system of equations. We can rewrite the above equation as two separate equations:
(400/9) x^2 - (364/9) x = 0
(1600/9) y^2 - (1722/9) y = 0
The first equation factors as:
(40/9) x (10x - 9) = 0
This means that x = 0 or x = 9/10.
The second equation factors as:
(800/9) y (2y - 21) = 0
This means that y = 0 or y = 21/2.
Since the points M, R, and P are collinear, they lie on the same line. Therefore, their y-coordinates must be the same. This means that y = 8 for all three points.
Therefore,
(x, y) = (9/10, 8) are the coordinates.