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Math  /  Geometry

QuestionTriangle ABCA B C is dilated to produce triangle A′B′C′A^{\prime} B^{\prime} C^{\prime}.
Determine the scale factor used to create the image. 13\frac{1}{3}

Studdy Solution

STEP 1

1. Triangle A′B′C′A'B'C' is the image of triangle ABCABC after dilation.
2. The dilation is centered at the origin or a point such that the coordinates of the vertices are scaled proportionally.
3. The coordinates of triangle ABCABC are A(3,0)A(3, 0), B(12,0)B(12, 0), C(3,12)C(3, 12).
4. The coordinates of triangle A′B′C′A'B'C' are A′(1,0)A'(1, 0), B′(4,0)B'(4, 0), C′(1,4)C'(1, 4).

STEP 2

1. Identify the corresponding vertices of the original and dilated triangles.
2. Calculate the scale factor using the coordinates of corresponding vertices.

STEP 3

Identify the corresponding vertices of the original and dilated triangles:
- AA corresponds to A′A' - BB corresponds to B′B' - CC corresponds to C′C'

STEP 4

Calculate the scale factor using the coordinates of corresponding vertices.
- For vertex AA to A′A': $ \text{Scale factor} = \frac{x_{A'}}{x_A} = \frac{1}{3} \]
- For vertex BB to B′B': $ \text{Scale factor} = \frac{x_{B'}}{x_B} = \frac{4}{12} = \frac{1}{3} \]
- For vertex CC to C′C': $ \text{Scale factor} = \frac{y_{C'}}{y_C} = \frac{4}{12} = \frac{1}{3} \]
Since all calculations yield the same scale factor, the scale factor is consistent.
The scale factor used to create the image is:
13 \boxed{\frac{1}{3}}

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