To solve this problem, we'll approach it step-by-step:

1. Rotation about point B(5,8):
   To rotate the triangle by a given angle about a point, we first translate the point to the origin, rotate, and then translate back. However, you haven't provided a specific rotation angle.

   If you meant a 90° counterclockwise rotation (a common choice), here's how we'd do it:

   The formula for a 90° counterclockwise rotation about a point $(x_0, y_0)$ is: $(x', y') = (y_0 - (y - y_0), x_0 + (x - x_0))$ So for each vertex, we will:

   1. Translate by subtracting point $B(5,8)$.
   2. Apply the 90° rotation.
   3. Translate back to point $B(5,8)$.

2. Scaling about point C(10,10):
   To scale by a factor of $(2,2)$ about point $C(10,10)$, we use the formula: $(x', y') = (C_x + s_x \cdot (x - C_x), C_y + s_y \cdot (y - C_y))$ Where $s_x = 2$ and $s_y = 2$ (scaling factors). We apply this scaling to each rotated vertex.

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Step 1: Rotate Triangle about Point $B(5,8)$

Let's first calculate the rotated positions (assuming a 90° counterclockwise rotation) of the vertices of the triangle about $B(5,8)$:

1. Vertex A(7, 15): $A' = (5 + (15 - 8), 8 - (7 - 5)) = (5 + 7, 8 - 2) = (12, 6)$

2. Vertex B(5, 8):
   This is the center of rotation, so it stays the same: $B' = (5, 8)$

3. Vertex C(10, 10): $C' = (5 + (10 - 8), 8 - (10 - 5)) = (5 + 2, 8 - 5) = (7, 3)$

So, the rotated triangle has vertices at $A'(12,6)$, $B'(5,8)$, and $C'(7,3)$.

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Step 2: Scale about Point $C(10,10)$

Now, we will scale the rotated triangle about $C(10,10)$ by a factor of $(2,2)$:

1. Vertex $A'(12, 6)$: $A'' = (10 + 2 \cdot (12 - 10), 10 + 2 \cdot (6 - 10)) = (10 + 4, 10 + (-8)) = (14, 2)$

2. Vertex $B'(5, 8)$: $B'' = (10 + 2 \cdot (5 - 10), 10 + 2 \cdot (8 - 10)) = (10 + (-10), 10 + (-4)) = (0, 6)$

3. Vertex $C'(7, 3)$: $C'' = (10 + 2 \cdot (7 - 10), 10 + 2 \cdot (3 - 10)) = (10 + (-6), 10 + (-14)) = (4, -4)$

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Final Answer:

The vertices of the triangle after a 90° counterclockwise rotation about $B(5,8)$ and scaling by $(2,2)$ about $C(10,10)$ are:

• $A''(14, 2)$
• $B''(0, 6)$
• $C''(4, -4)$

Would you like more details about how to perform these transformations, or do you have any specific questions?

Follow-up Questions:

1. How do rotation transformations work in 2D?
2. What happens if the scaling factor is not uniform (different for $x$ and $y$)?
3. How can you apply a rotation for an arbitrary angle?
4. What is the effect of scaling by a factor less than 1?
5. How do translation and rotation interact geometrically?

Tip:

When performing transformations, always work step by step (translate, rotate, then translate back) for clarity and accuracy.