Let P be the point (0,-1) and Q be the point (3,2). In this Cartesian coordinate system, find the point on the line segment PQ that is twice as far from P as from Q.
Please help, I need this for a test and i have totally forgotten the formula and notes
and how to do anything regarding this.
I am sure you made a sketch
Let M(x,y) be that point, so that PM:MQ = 2:1
complete the two right-angled triangles with PM and PQ as the hypotenuse.
by ratios:
(x-0)/(3-0) = 2/3
3x=6
x=2
(y+1)/(2+1)=2/3
3y+3=6
3y=3
y=1
your point is (2,1)
check:
PM = sqrt(2^2+2^2)= sqrt(8) = 2sqrt(2)
MQ = sqrt(1^2 + 1^2) = sqrt(2)
thus PM = 2MQ , as required
Ah, don't worry! I'm here to help, and I promise it won't be as scary as facing a clown with a balloon sword!
To find the point on the line segment PQ that is twice as far from P as from Q, we can break it down into a few steps.
Step 1: Calculate the distance between P and Q.
We can use the distance formula, which states that the distance between two points (x1, y1) and (x2, y2) is the square root of ((x2 - x1)^2 + (y2 - y1)^2).
So, using the given points P(0, -1) and Q(3, 2), the distance between P and Q is:
d(PQ) = sqrt((3 - 0)^2 + (2 - (-1))^2)
= sqrt(3^2 + 3^2)
= sqrt(9 + 9)
= sqrt(18)
= 3sqrt(2)
Step 2: Determine the point on PQ that is twice as far from P as from Q.
Let's call this point R. If R is twice as far from P as from Q, then the ratio of the distance between P and R to the distance between R and Q should be 2:1.
Let's say the coordinates of point R are (x, y). To find these coordinates, we'll use the midpoint formula, which states that the midpoint of a line segment with endpoints (x1, y1) and (x2, y2) is ((x1 + x2)/2, (y1 + y2)/2).
So, the coordinates of the midpoint R are:
x = (0 + 3)/2
= 3/2
= 1.5
y = (-1 + 2)/2
= 1/2
= 0.5
Therefore, the point R on the line segment PQ that is twice as far from P as from Q is R(1.5, 0.5).
I hope that helps! Now go ace that test and have a laugh along the way!
To find the point on the line segment PQ that is twice as far from P as from Q, we can use the concept of the midpoint formula. We know that the point we are looking for is twice as far from P as from Q, which means the ratio of the distances is 2:1.
Here are the steps to find the desired point:
1. Calculate the distance between points P and Q using the distance formula:
Distance = √[(x2 - x1)^2 + (y2 - y1)^2]
Plugging in the coordinates of P and Q:
Distance = √[(3 - 0)^2 + (2 - (-1))^2]
= √[9 + 9]
= √18
≈ 4.24
2. Divide the distance between P and Q by 3 to determine one-third of the distance:
One-third distance = (√18) / 3
≈ 1.41
3. Determine the coordinates of the point that is one-third of the distance from Q to P:
- Starting from point Q, move one-third of the distance towards P.
- Since the x-coordinate of P is 0 and the x-coordinate of Q is 3, we will subtract (1.41) from 3 to get the x-coordinate of the desired point: x = 3 - 1.41 = 1.59
- Since the y-coordinate of P is -1 and the y-coordinate of Q is 2, we will subtract (1.41) from 2 to get the y-coordinate of the desired point: y = 2 - 1.41 = 0.59
Therefore, the point on the line segment PQ that is twice as far from P as from Q is approximately (1.59, 0.59).
To find the point on the line segment PQ that is twice as far from P as from Q, we can use the concept of the midpoint formula and the concept of section formula.
Let R(x, y) be the point we are looking for.
First, let's find the coordinates of the midpoint of line segment PQ. The midpoint is simply the average of the x-coordinates and the average of the y-coordinates of points P and Q.
Midpoint formula:
Midpoint M = ((x1 + x2) / 2, (y1 + y2) / 2)
For points P(0,-1) and Q(3,2):
Midpoint M = ((0 + 3) / 2, (-1 + 2) / 2)
= (3/2, 1/2)
= (1.5, 0.5)
Now, let's use the section formula to find the coordinates of point R(x, y), which is twice as far from P as from Q.
Section formula:
If a point R divides a line segment PQ in the ratio m:n, then the coordinates of R are given by:
x = (mx2 + nx1) / (m + n)
y = (my2 + ny1) / (m + n)
In this case, we want R to be twice as far from P as from Q. This means the ratio is 2:1.
Using the section formula for points P(0,-1) and Q(3,2):
x = (2 * 3 + 1 * 0) / (2 + 1)
= (6 + 0) / 3
= 6 / 3
= 2
y = (2 * 2 + 1 * (-1)) / (2 + 1)
= (4 - 1) / 3
= 3 / 3
= 1
Therefore, the coordinates of point R on line segment PQ that is twice as far from P as from Q are (2, 1).