Point P is located on the segment between point A(1,4) and point D(7,13). The distance from A to P is twice the distance from P to D. What are the coordinates of point P?
PD = d,
AP = 2d,
AP/AD = 2d/(d+2d) = 2d/3d = 2/3.
AP/AD = 2/3,
AP = (2/3)AD.
x -1 = @/3(7-1),
x -1 = 4,
X = 4+1 = 5.
y-4 = 2/3(13-4),
y-4 = 6,
Y = 6 + 4 = 10.
P(5,10).
Correction: x-1 = 2/3(7-1).
To find the coordinates of point P, we can use the concept of the midpoint formula.
Let the coordinates of point P be (x, y).
According to the question, the distance from A to P is twice the distance from P to D.
Using the distance formula, we can write the equation as follows:
√[(x - 1)^2 + (y - 4)^2] = 2 * √[(x - 7)^2 + (y - 13)^2]
Squaring both sides of the equation to remove the square roots, we get:
(x - 1)^2 + (y - 4)^2 = 4 * [(x - 7)^2 + (y - 13)^2]
Expanding and simplifying the equation, we have:
x^2 - 2x + 1 + y^2 - 8y + 16 = 4 * (x^2 - 14x + 49 + y^2 - 26y + 169)
Simplifying further, we get:
x^2 - 2x + y^2 - 8y + 17 = 4x^2 - 56x + 196 + 4y^2 - 104y + 676
Combining like terms, we obtain:
3x^2 - 54x + 3y^2 - 96y + 483 = 0
Dividing the equation by 3, we get:
x^2 - 18x + y^2 - 32y + 161 = 0
To find the coordinates of point P, we need to solve this equation. However, it seems that there is not enough information provided in the question to determine the specific coordinates of point P. If there are additional conditions given or if some information is missing, please provide it so that we can continue with the calculations.
To find the coordinates of point P, we need to use the given information that the distance from A to P is twice the distance from P to D.
Let's start by finding the distance between points A(1,4) and D(7,13). We can use the distance formula:
Distance = sqrt((x2 - x1)^2 + (y2 - y1)^2)
In this case, x1 = 1, y1 = 4, x2 = 7, and y2 = 13.
Distance AD = sqrt((7 - 1)^2 + (13 - 4)^2)
= sqrt(6^2 + 9^2)
= sqrt(36 + 81)
= sqrt(117)
Now, according to the given information, the distance from A to P is twice the distance from P to D. Let's denote the distance from A to P as 2x, and the distance from P to D as x.
So, the distance AP would be 2x, and the distance PD would be x.
We can set up the equation: 2x + x = sqrt(117)
Combining like terms, we get: 3x = sqrt(117)
Squaring both sides of the equation yields: (3x)^2 = (sqrt(117))^2
Simplifying, we have: 9x^2 = 117
Dividing both sides by 9, we get: x^2 = 13
Taking the square root of both sides gives us: x = sqrt(13)
Now, we know that the distance from P to D is x=sqrt(13). To find the coordinates of point P, we need to move x units from D towards A.
The x-coordinate of D is 7, so the x-coordinate of point P would be: 7 - sqrt(13)
Similarly, the y-coordinate of D is 13, so the y-coordinate of point P would be: 13 - sqrt(13)
Thus, the coordinates of point P are (7 - sqrt(13), 13 - sqrt(13)).