please illustrate how to rotate a triangle counterclockwise from the origin with coordinates A(2,0),B(7,0),
C(7,2). Thank you.
how can i help my son matthew about division and subtract borrow . thanks
Please click Post a New Question and then ask your question. Also -- your question isn't clear. What exactly does Matthew need to know about division? Do you also want to know about how to borrow in subtraction problems?
http://www.aaastudy.com/div.htm
http://www.aaastudy.com/sub.htm
For each point to be rotated, use this set of equations:
x'=x*cos(theta)+y*sin(theta)
y'=-x*sin(theta)+y*cos(theta)
where x,y is the original point, and theta is the chosen angle of rotation counterclockwise.
To rotate a triangle counterclockwise from the origin, you can follow these steps:
1. Plot the triangle points A(2, 0), B(7, 0), and C(7, 2) on a coordinate plane.
2. Find the coordinates of the origin, which is (0, 0).
3. Draw a line connecting each triangle point with the origin.
4. Measure the angle formed between the line from the origin to point A and the positive x-axis. Let's call this angle "θ".
5. To rotate counterclockwise, you need to change each point's coordinates using the rotation matrix:
- The new x-coordinate of a point after rotation is given by: x' = x * cos(θ) - y * sin(θ)
- The new y-coordinate of a point after rotation is given by: y' = x * sin(θ) + y * cos(θ)
6. Apply the rotation matrix to each point of the triangle:
- For point A(2, 0):
- x' = 2 * cos(θ) - 0 * sin(θ) = 2 * cos(θ)
- y' = 2 * sin(θ) + 0 * cos(θ) = 2 * sin(θ)
- For point B(7, 0):
- x' = 7 * cos(θ) - 0 * sin(θ) = 7 * cos(θ)
- y' = 7 * sin(θ) + 0 * cos(θ) = 7 * sin(θ)
- For point C(7, 2):
- x' = 7 * cos(θ) - 2 * sin(θ)
- y' = 7 * sin(θ) + 2 * cos(θ)
7. Substitute the value of θ based on the angle of rotation you desire. Let's say you want to rotate counterclockwise by 45 degrees, which is equal to π/4 radians. Therefore, θ = π/4.
8. Calculate the new coordinates for each point:
- For point A(2, 0):
- x' = 2 * cos(π/4) = 2 * √2/2 = √2
- y' = 2 * sin(π/4) = 2 * √2/2 = √2
- For point B(7, 0):
- x' = 7 * cos(π/4) = 7 * √2/2 = (7√2)/2
- y' = 7 * sin(π/4) = 7 * √2/2 = (7√2)/2
- For point C(7, 2):
- x' = 7 * cos(π/4) - 2 * sin(π/4) = (7√2)/2 - 2(√2)/2 = (5√2)/2
- y' = 7 * sin(π/4) + 2 * cos(π/4) = (7√2)/2 + 2(√2)/2 = (11√2)/2
9. Plot the transformed triangle with the new coordinates (A'(√2, √2), B'((7√2)/2, (7√2)/2), C'((5√2)/2, (11√2)/2)) on the same coordinate plane.
Please note that in step 8, I have rounded the results for simplicity, but you can keep the exact values if needed.