A triangle T with vertices A (2,4), B(6,2), and C(4,8) is mapped onto Triangle T’ with vertices A’(10,0), B’(8,-4) and C’(14,-2) by a rotation.
Determine the center and angle of rotation that maps T onto T’.
To find the center and angle of rotation that maps triangle T onto T', we will use the following steps:
1. Find the midpoints of two corresponding sides of each triangle.
2. Determine the translation vector between the midpoints.
3. Find the length of the corresponding sides.
4. Calculate the cosine of the rotation angle using the dot product.
5. Determine the rotation angle.
6. Calculate the center of rotation.
Let's go through each step in detail:
Step 1: Find the midpoints of two corresponding sides of each triangle.
The midpoint of side AB in triangle T is:
X_AB = ( (x1 + x2) / 2 , (y1 + y2) / 2)
= ( (2 + 6) / 2 , (4 + 2) / 2)
= ( 8 / 2 , 6 / 2)
= (4, 3)
The midpoint of side A'B' in triangle T' is:
X_A'B' = ((x1' + x2') / 2 , (y1' + y2') / 2)
= ((10 + 8) / 2 , (0 + (-4)) / 2)
= (18 / 2 , -4 / 2)
= (9, -2)
Similarly, find the midpoints of side AC and side A'C'.
The midpoint of side AC in triangle T is:
X_AC = ((x1 + x2) / 2 , (y1 + y2) / 2)
= ((2 + 4) / 2 , (4 + 8) / 2)
= (6 / 2 , 12 / 2)
= (3, 6)
The midpoint of side A'C' in triangle T' is:
X_A'C' = ((x1' + x2') / 2 , (y1' + y2') / 2)
= ((10 + 14) / 2 , (0 + (-2)) / 2)
= (24 / 2 , -2 / 2)
= (12, -1)
Step 2: Determine the translation vector between the midpoints.
Translation vector, V = X_A'C' - X_AC
= (12, -1) - (3, 6)
= (9, -7)
Step 3: Find the length of the corresponding sides.
The length of side AB is:
|AB| = sqrt((x2 - x1)^2 + (y2 - y1)^2)
= sqrt((6 - 2)^2 + (2 - 4)^2)
= sqrt(4^2 + (-2)^2)
= sqrt(16 + 4)
= sqrt(20)
= 2*sqrt(5)
The length of side A'B' is:
|A'B'| = sqrt((x2' - x1')^2 + (y2' - y1')^2)
= sqrt((8 - 10)^2 + (-4 - 0)^2)
= sqrt((-2)^2 + (-4)^2)
= sqrt(4 + 16)
= sqrt(20)
= 2*sqrt(5)
Step 4: Calculate the cosine of the rotation angle using the dot product.
Cosine of the rotation angle, cos(theta) = (V • AB) / (|V| * |AB|)
= (9*4 + (-7)*(-2)) / (sqrt(9^2 + (-7)^2) * sqrt(20))
= (36 + 14) / (sqrt(81 + 49) * (2*sqrt(5))
= 50 / (sqrt(130) * 2 * sqrt(5))
= 25 / (2 * sqrt(130))
Step 5: Determine the rotation angle.
The rotation angle, theta = arccos(cos(theta))
= arccos(25 / (2 * sqrt(130)))
Step 6: Calculate the center of rotation.
The center of rotation is the translation vector divided by the sine of the rotation angle.
Center of rotation, C = V / sin(theta)
= (9, -7) / sin(arccos(25 / (2 * sqrt(130))))
Therefore, to determine the center and angle of rotation that maps triangle T onto T':
Center of rotation = C = (9, -7)
Angle of rotation = theta = arccos(25 / (2 * sqrt(130)))
To determine the center and angle of rotation that maps triangle T onto T', we can use the following steps:
1. Find the midpoints of triangle T and T':
- The midpoint of T can be found by averaging the coordinates of its vertices:
Midpoint of T = ((2+6+4)/3, (4+2+8)/3) = (4, 4.67)
- The midpoint of T' can be found similarly:
Midpoint of T' = ((10+8+14)/3, (0-4-2)/3) = (10.67, -2)
2. Determine the translation vector from the midpoints:
- The translation vector can be obtained by subtracting the midpoint of T from the midpoint of T':
Translation vector = Midpoint of T' - Midpoint of T = (10.67-4, -2-4.67) = (6.67, -6.67)
3. Apply the translation vector to any point on triangle T to get the corresponding point on T':
- Let's use point A (2,4) as an example:
Point A' = Point A + Translation vector = (2,4) + (6.67, -6.67) = (8.67, -2.67)
4. Find the center of rotation:
- The center of rotation is the image of the origin (0, 0) after applying the translation vector:
Center of rotation = Origin + Translation vector = (0,0) + (6.67, -6.67) = (6.67, -6.67)
5. Determine the angle of rotation:
- To find the angle of rotation, we can use the relationship between the vectors formed by the corresponding points on triangles T and T':
- Let's use points A (2,4) and A' (8.67, -2.67) as a reference:
- Vector T = (2,4) - (4,4.67) = (-2, -0.67)
- Vector T' = (8.67, -2.67) - (6.67, -6.67) = (2, 4)
- The angle of rotation can be found using the dot product formula:
Angle of rotation = arccos((Vector T • Vector T') / (|Vector T| * |Vector T'|))
= arccos(((-2)(2) + (-0.67)(4)) / (√((-2)^2 + (-0.67)^2) * √(2^2 + 4^2)))
≈ arccos(-0.38)
≈ 113.19° (rounded to two decimal places)
Therefore, the center of rotation that maps triangle T onto T' is (6.67, -6.67) and the angle of rotation is approximately 113.19°.