three points have the coordinates A(2,9) B(4,3) and C(2,-5). The line through C with gradient 0.5 meets the line AB produced at D. Calculate the coordinates of D.
Using the two-point form of the line, line AB is
y-9 = (3-9)/(4-2) * (x-2)
y = -3x + 15
Using the point-slope form of the line, line CD is
y-(-5) = .5 (x-2)
y = .5x - 6
So, point D is where the two lines intersect:
-3x + 15 = .5x - 6
21 = 3.5x
x = 6
Since CD has slope .5, y increases by .5 when x increases by 1. Since x=6 is 4 units to the right, we want y 2 more than -5 = -3
So, D = (6,-3)
Well, I'm not really a math whiz, but I can certainly try to help you out with some humor!
Okay, let's see... The line through C with a gradient of 0.5. So, C is basically saying, "Hey, I want to have a nice, gentle slope!"
And then you have line AB, which is probably like, "Nah, I'm gonna do my own thing, who needs to follow a gradient?"
But then, D is like, the meeting point of these two rebellious lines, where AB finally grudgingly agrees to cross paths with C.
So, to find the coordinates of D, let's imagine that AB and C are characters in a comedy skit. They're standing on a stage, trying to figure out where they intersect.
Now, since C has a gradient of 0.5, it means that for every 1 unit it moves horizontally, it moves 0.5 units vertically. It's like C is doing a half-hearted attempt at salsa dancing!
So, to find where C intersects with AB, we need to find the point where they both agree to get together. And that point is D!
Now, to calculate the coordinates of D, we need to figure out where AB is when C finally catches up. Since C is moving from left to right on the stage, let's keep that in mind.
The x-coordinate of D will be the same as C because they intersect vertically, and since C is at x = 2, that's the x-coordinate of D!
To find the y-coordinate of D, we need to figure out how far AB has moved vertically when C catches up. But since AB is doing its own thing and not following any gradient, let's just consider the difference in the y-values of A and B.
The y-coordinate of D will be the y-coordinate of B plus the difference in the y-values of A and B. In other words, it's like B telling A, "Hey, I'm going to add a bit of flavor to our meeting point!"
So, the y-coordinate of D is 3 (the y-coordinate of B) plus 9 (the y-coordinate of A) minus 3 (the y-coordinate of B), which is 9.
Therefore, the coordinates of D are (2, 9). Ta-da!
I hope my little comedy skit helped you visualize the problem and find the solution!
To find the coordinates of point D, we need to find the equation of the line passing through point C with a gradient of 0.5.
First, let's find the slope (gradient) of the line passing through points A and B. We can use the slope formula:
m = (y2 - y1) / (x2 - x1)
Where (x1, y1) = (2, 9) and (x2, y2) = (4, 3).
m = (3 - 9) / (4 - 2)
m = -6 / 2
m = -3
So the slope of line AB is -3.
Now we can find the equation of line AB using the point-slope form:
y - y1 = m(x - x1)
Using point A (2, 9), we have:
y - 9 = -3(x - 2)
y - 9 = -3x + 6
y = -3x + 15
Now, let's find the equation of the line passing through point C (2, -5) with a gradient of 0.5 using point-slope form:
y - y1 = m(x - x1)
Using point C (2, -5) and m = 0.5, we have:
y - (-5) = 0.5(x - 2)
y + 5 = 0.5x - 1
y = 0.5x - 6
Now, let's solve the two equations:
-3x + 15 = 0.5x - 6
Add 3x and subtract 0.5x from both sides:
15 + 6 = 3.5x
21 = 3.5x
Divide by 3.5:
x = 6
Now substitute x = 6 into either equation to find y:
y = 0.5(6) - 6
y = 3 - 6
y = -3
Therefore, the coordinates of point D are (6, -3).
To calculate the coordinates of point D, we need to find the equation of the line passing through point C with a gradient of 0.5.
Step 1: Find the equation of the line through C with gradient 0.5
The gradient-intercept form of a linear equation is y = mx + c, where m represents the gradient and c represents the y-intercept.
Given that the gradient of the line passing through point C is 0.5, we can write the equation as:
y = 0.5x + c
To find the value of c, we substitute the coordinates of point C (2, -5) into the equation:
-5 = 0.5(2) + c
-5 = 1 + c
c = -6
The equation of the line through point C with a gradient of 0.5 is:
y = 0.5x - 6
Step 2: Find the equation of the line AB
Given the coordinates of points A(2,9) and B(4,3), we can find the equation of line AB using the slope-intercept form.
The slope (m) of line AB can be found using the formula:
m = (y2 - y1) / (x2 - x1)
Substituting the coordinates of A and B into the formula:
m = (3 - 9) / (4 - 2)
m = -6 / 2
m = -3
Now, we can write the equation of line AB using the point-slope form and the coordinates of one of the points, for example, point A(2,9):
y - 9 = -3(x - 2)
y - 9 = -3x + 6
y = -3x + 15
Step 3: Find the coordinates of point D
Point D is the intersection of the line through point C with gradient 0.5, and the line AB.
To find the coordinates of point D, we need to solve the equations of both lines simultaneously.
Substituting the equation of line AB into the equation of the line through point C:
-3x + 15 = 0.5x - 6
Simplifying the equation:
-3x - 0.5x = -6 - 15
-3.5x = -21
x = -21 / -3.5
x = 6
Substituting the value of x into either equation, let's use the equation of line AB:
y = -3(6) + 15
y = -18 + 15
y = -3
Therefore, the coordinates of point D are D(6, -3).