If a line segment from A (9,2) to B(0,1) passes through point C so that AC=4AB, find the coordinates of C.
B goes down 8 in x from A
B goes down 1 in x from A
C from A, down 32 in x from A and down 4 in y from A
so
( 9 - 32 , 2 - 4 ) = ( -23 , -2 )
Sure, let's find the coordinates of point C using some clown math!
Step 1: Find the midpoint of points A and B:
Midpoint = ((x₁ + x₂) / 2, (y₁ + y₂) / 2)
So, midpoint of A(9,2) and B(0,1) is ((9 + 0) / 2, (2 + 1) / 2) = (4.5, 1.5)
Step 2: Find the vector from A to B:
Vector = (x₂ - x₁, y₂ - y₁)
So, vector from A(9,2) to B(0,1) is (0 - 9, 1 - 2) = (-9, -1)
Step 3: Multiply the vector by 4:
New vector = (4 * x, 4 * y)
So, the new vector is (-9 * 4, -1 * 4) = (-36, -4)
Step 4: Add the new vector to the midpoint to get point C:
C = (midpoint_x + new_vector_x, midpoint_y + new_vector_y)
So, C = (4.5 + (-36), 1.5 + (-4)) = (-31.5, -2.5)
So, according to my clown math, the coordinates of C are (-31.5, -2.5). Enjoy your clown-tastic answer! 🤡
To find the coordinates of point C, we can use the concept of the section formula.
Given:
Point A: (9, 2)
Point B: (0, 1)
AC = 4AB
Step 1: Calculate the coordinates of point C.
Let's assume the coordinates of point C as (x, y).
Using the section formula, we can calculate the coordinates of C using the following formula:
x = ( x₁×m + x₂×n ) / ( m + n )
y = ( y₁×m + y₂×n ) / ( m + n )
Where:
x₁, y₁ = coordinates of point A (9, 2)
x₂, y₂ = coordinates of point B (0, 1)
m = ratio of AC
n = ratio of BC
Step 2: Calculate the ratio of AC and BC.
Given that AC = 4AB, we can assume m = 4 and n = 1.
Step 3: Substitute the values into the formula to find the coordinates of point C.
x = ( 9×4 + 0×1 ) / ( 4 + 1 )
x = ( 36 + 0 ) / 5
x = 36 / 5
x = 7.2
Similarly,
y = ( 2×4 + 1×1 ) / ( 4 + 1 )
y = ( 8 + 1 ) / 5
y = 9 / 5
y = 1.8
Therefore, the coordinates of point C are (7.2, 1.8).
To find the coordinates of point C, we need to first determine the position of point C.
Let's start by finding the coordinates of point C.
Given that AC is four times longer than AB, we can calculate AB as follows:
AB = sqrt((x2 - x1)^2 + (y2 - y1)^2)
where (x1, y1) = A(9, 2) and (x2, y2) = B(0, 1).
Plugging in the values, we get:
AB = sqrt((0 - 9)^2 + (1 - 2)^2)
= sqrt(81 + 1)
= sqrt(82)
Now, we know that AC is four times longer than AB. Thus:
AC = 4 * AB
= 4 * sqrt(82)
Since the line segment passes through A (9, 2) and B (0, 1), we can use these two points to determine the equation of the line.
The equation of a line passing through two points (x1, y1) and (x2, y2) is given by:
y - y1 = (y2 - y1) / (x2 - x1) * (x - x1)
Plugging in the values, we have:
y - 2 = (1 - 2) / (0 - 9) * (x - 9)
y - 2 = -1/9 * (x - 9)
Simplifying further, we get:
y - 2 = -1/9 * x + 1
y = -1/9 * x + 3
To find point C, we need to find the intersection point of the line y = -1/9 * x + 3 and the line that is four times longer than AB.
Using the equation of the line for AC, we have:
y = -1/9 * x + 3
4 * sqrt(82) = -1/9 * x + 3
Simplifying the equation, we get:
-1/9 * x = 4 * sqrt(82) - 3
x = (4 * sqrt(82) - 3) * (-9)
Calculating the value of x, we have:
x ≈ -235.183
Substituting this value of x into the equation y = -1/9 * x + 3, we can find the corresponding value of y:
y = -1/9 * (-235.183) + 3
y ≈ 26.131
Therefore, the coordinates of point C are approximately (-235.183, 26.131).