Let P be (-3,7) and Q be (5,-12)
Find the point T on line segment PQ such that PT/TQ = 1/3
How would I go about solving this problem?
Thank you so much!
let the point be T(x,y)
then (x+3)/(5-x) = 1/3
3x+9 = 5-x
x = -1
do the same for the y
(y-7)/(-12-y) = 1/3
3y-21 = -12-y
y = 9/4
Could you try to explain it a little more?
I was simply using your given ratio.
(change in x's from P to T)/(change in x's from T to Q) = 1/3
do the same for the y's
You can use this method to divide a line segment into any given ratio.
e endpoints of $\overline{AB}$ are $A(-4,\ -8)$ and $B(-1,\ 4)$ . Find the coordinates of the midpoint $M$ .
The coordinates of the midpoint $M$ are (
,
)
e endpoints of $\overline{AB}$ are $A(-4,\ -8)$ and $B(-1,\ 4)$ . Find the coordinates of the midpoint $M$ .
The coordinates of the midpoint $M$ are (
,
)
Well, solving a problem involving points and line segments sounds quite "pointless," doesn't it?
But fear not, my friend! I am here to bring some clowny logic into the equation.
To find point T on line segment PQ such that PT/TQ = 1/3, we can start by finding the coordinates of point T.
Let's use a little math magic. First, we need to find the distance between points P and Q. We can use the distance formula:
d = √[(x₂ - x₁)² + (y₂ - y₁)²]
In this case, the distance between P and Q is:
d = √[(5 - (-3))² + (-12 - 7)²]
After some number-crunching, we get the following distance:
d = √[64 + 361]
Now that we have the total distance, we can break it down into a ratio. The sum of the ratio parts is 1 + 3 = 4. So, point T is 1/4th of the total distance from point P to point Q.
Let's calculate the coordinates of T:
x-coordinate of T = (1/4)(5) + (3/4)(-3)
y-coordinate of T = (1/4)(-12) + (3/4)(7)
After some more math gymnastics, we find that:
x-coordinate of T = -1
y-coordinate of T = 4
So, the point T on line segment PQ, such that PT/TQ = 1/3, is (-1,4).
Now that wasn't too "pointless," was it?
To find the point T on the line segment PQ such that PT/TQ = 1/3, you can start by calculating the coordinates of point T using the concept of section formula.
The section formula states that if two points A and B divide a line segment in a given ratio, say m:n, then the coordinates of the point dividing the line segment internally in this ratio are given by:
Px = (n * Ax + m * Bx) / (m + n)
Py = (n * Ay + m * By) / (m + n)
In this case, we want to find the coordinates of point T, which divides the line segment PQ in the ratio 1:3. Therefore, m = 1 and n = 3.
Let's calculate the coordinates of T using the section formula:
PT = 1/3 of PQ
PT = (1 * Px + 3 * Qx) / (1 + 3)
PT = (1 * (-3) + 3 * 5) / 4
PT = (-3 + 15) / 4
PT = 12 / 4
PT = 3
Applying the same calculation for the y-coordinates:
PT = (1 * Py + 3 * Qy) / (1 + 3)
PT = (1 * 7 + 3 * (-12)) / 4
PT = (7 - 36) / 4
PT = -29 / 4
Therefore, the coordinates of point T are (3, -29/4) or (3, -7.25).
Thus, the point T on the line segment PQ, such that PT/TQ = 1/3, is (3, -7.25).