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?
Just need to know how to solve but don't just leave an answer because there are 3 more problems likewise.
simplest way is to use ratios
for the x,
(x-1)/(7-x) = 2/1
x-1 = 14-2x
3x = 15
x = 5
for the y,
(y-1)/(13-y) = 2/1
y-1 = 26-2y
3y = 27
y = 9
so point P is (5,9)
check:
AP = √(5-1)^2 + (9-1)^2
= √(16+64) = √80
= 4√5
PD = √((7-5)^2 + (13-9)^2)
= √(4 + 16) = √20
= 2√5
4√5 : 2√5 = 2 : 1
Do your other questions the same way
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 first find the distance between two points using the distance formula:
The distance between two points (x1, y1) and (x2, y2) is given by the formula:
d = sqrt((x2 - x1)^2 + (y2 - y1)^2)
Let's consider point P to have coordinates (x, y).
Using the distance formula, we can set up the following equations based on the given information:
Distance from A to P:
d(A, P) = sqrt((x - 1)^2 + (y - 4)^2)
Distance from P to D:
d(P, D) = sqrt((7 - x)^2 + (13 - y)^2)
According to the problem, the distance from A to P is twice the distance from P to D, so we can set up the equation:
2 * d(P, D) = d(A, P)
Squaring both sides of the equation gives us:
4 * (d(P, D))^2 = (d(A, P))^2
Substituting in the distance formula expressions, we have:
4 * ((7 - x)^2 + (13 - y)^2) = (x - 1)^2 + (y - 4)^2
Expand and simplify:
4 * (49 - 14x + x^2 + 169 - 26y + y^2) = x^2 - 2x + 1 + y^2 - 8y + 16
Simplify further:
196 - 56x + 4x^2 + 676 - 104y + 4y^2 = x^2 - 2x + 1 + y^2 - 8y + 16
Rearranging the terms and combining like terms:
3x^2 + 2x + 3y^2 - 96y + 443 = 0
Now, we have a quadratic equation in terms of x and y. To solve for the coordinates of point P, we need to find the values of x and y that satisfy this equation.
Solving this quadratic equation would involve applying appropriate methods like factoring, completing the square, or using the quadratic formula. By solving for either x or y in terms of the other variable, we can substitute back into one of the original distance formula equations to find the other variable.
Unfortunately, without further information or constraints, it is not possible to determine the specific coordinates of point P.