Trapezoid ABCD

Question

has vertices A(–5
, 10)
, B(10
, 10)
, C(5
, 5)
, and D(0
, 5)
. Graph the trapezoid and its image after a dilation with a scale factor of 0.2.

Multiple choice question.

Trapezoid A B C D and Trapezoid A prime (negative 1, negative 2), B prime (2, negative 2), C prime (1, negative 1), D prime (0, negative 1).

Trapezoid A B C D and Trapezoid A prime (negative 2, negative 2), B prime (4, negative 2), C prime (2, negative 1), D prime (0, negative 1).

Trapezoid A B C D and Trapezoid A prime (negative 2, 4), B prime (4, 4), C prime (2, 2), D prime (0, 2).

Trapezoid A B C D and Trapezoid A prime (negative 1, 2), B prime (2, 2), C prime (1, 1), D prime (0, 1).

Answer 1

To find the image of trapezoid ABCD after a dilation with a scale factor of 0.2, we first need to determine the center of dilation.

Assuming the center of dilation is at the origin (0,0), the formula for a dilation of a point \((x, y)\) with respect to a scale factor \(k\) is given by: \[ (x', y') = (kx, ky) \]

Now let's apply the dilation with a scale factor of 0.2 to each vertex of trapezoid ABCD:

1. For point A(–5, 10): \[ A' = (0.2 \times -5, 0.2 \times 10) = (-1, 2) \]

2. For point B(10, 10): \[ B' = (0.2 \times 10, 0.2 \times 10) = (2, 2) \]

3. For point C(5, 5): \[ C' = (0.2 \times 5, 0.2 \times 5) = (1, 1) \]

4. For point D(0, 5): \[ D' = (0.2 \times 0, 0.2 \times 5) = (0, 1) \]

After performing the dilation, the coordinates of the image trapezoid A'B'C'D' are:

• A'(-1, 2)
• B'(2, 2)
• C'(1, 1)
• D'(0, 1)

Now let’s find the correct option among the provided choices:

1. Option 1: A'(-1, -2), B'(2, -2), C'(1, -1), D'(0, -1) - Incorrect
2. Option 2: A'(-2, -2), B'(4, -2), C'(2, -1), D'(0, -1) - Incorrect
3. Option 3: A'(-2, 4), B'(4, 4), C'(2, 2), D'(0, 2) - Incorrect
4. Option 4: A'(-1, 2), B'(2, 2), C'(1, 1), D'(0, 1) - Correct

Thus, the correct answer is Option 4: Trapezoid A B C D and Trapezoid A'(-1, 2), B'(2, 2), C'(1, 1), D'(0, 1).

Answer 2

Triangle ABC
has vertices A(–18,0)
, B(–3, –3)
, and C(0, –12)
. Graph the triangle and its image after a dilation with a scale factor of 16
.

Multiple choice question.

Triangles A B C and A prime B prime C prime. A prime (negative 5.5, negative 2.5), B prime negative (3, negative 3), C prime (negative 2.5, negative 5).

Triangles A B C and A prime B prime C prime. A prime (negative 6, 0), B prime (negative 3.5, negative 0.5), C prime (negative 3, negative 2).

Triangles A B C and A prime B prime C prime. A prime (negative 3, 0), B prime (negative 0.5, negative 0.5), C prime (0, negative 2).

Triangles A B C and A prime B prime C prime. A prime (negative 6, 0), B prime (negative 1, negative 1), C prime (0, negative 4).

Answer 3

To find the image of triangle ABC after a dilation with a scale factor of 16, we need to determine the coordinates of the new points based on a center of dilation, which we will assume to be the origin (0,0) unless stated otherwise.

The formula for a dilation of a point \((x, y)\) with respect to a scale factor \(k\) is given by: \[ (x', y') = (kx, ky) \]

Now, let’s apply the dilation with a scale factor of 16 to each vertex of triangle ABC:

1. For point A(–18, 0): \[ A' = (16 \times -18, 16 \times 0) = (-288, 0) \]

2. For point B(–3, –3): \[ B' = (16 \times -3, 16 \times -3) = (-48, -48) \]

3. For point C(0, –12): \[ C' = (16 \times 0, 16 \times -12) = (0, -192) \]

After calculating the new vertex positions, we have:

• A'(-288, 0)
• B'(-48, -48)
• C'(0, -192)

Now we can review the options given:

1. Option 1: A'(-5.5, -2.5), B'(-3, -3), C'(-2.5, -5) - Incorrect
2. Option 2: A'(-6, 0), B'(-3.5, -0.5), C'(-3, -2) - Incorrect
3. Option 3: A'(-3, 0), B'(-0.5, -0.5), C'(0, -2) - Incorrect
4. Option 4: A'(-6, 0), B'(-1, -1), C'(0, -4) - Incorrect

None of the options provided match the results obtained from the dilation of triangle ABC with a scale factor of 16 and center at the origin. If there are any additional details about the center of dilation, or if I should assume a different center, please let me know! Otherwise, the dilation results clearly indicate that none of the given answer choices are correct based on my calculations.

Answer 4

A square with 3 foot long sides is dilated by a scale factor of 4. What are the dimensions of the square after the dilation?

Answer 5

To find the dimensions of the square after dilation, we multiply the original side length of the square by the scale factor.

The original side length of the square is 3 feet. The scale factor is 4.

Calculating the new side length after dilation:

\[ \text{New Side Length} = \text{Original Side Length} \times \text{Scale Factor} = 3 , \text{feet} \times 4 = 12 , \text{feet} \]

Thus, after the dilation, the dimensions of the square will be 12 feet on each side.

Answer 6

A triangle has sides measuring 6, 12, and 4 is dilated to a triangle with sides measuring 9, 18, and 6. What is the scale factor of the dilation?

Answer 7

To find the scale factor of the dilation, we can compare the lengths of corresponding sides of the original triangle and the dilated triangle.

The sides of the original triangle are \(6\), \(12\), and \(4\).

The sides of the dilated triangle are \(9\), \(18\), and \(6\).

We can calculate the scale factor by dividing the lengths of the sides of the dilated triangle by the lengths of the corresponding sides of the original triangle.

Let's check each pair of corresponding sides:

1. For the sides measuring \(6\) and \(9\): \[ \text{Scale Factor} = \frac{9}{6} = 1.5 \]

2. For the sides measuring \(12\) and \(18\): \[ \text{Scale Factor} = \frac{18}{12} = 1.5 \]

3. For the sides measuring \(4\) and \(6\): \[ \text{Scale Factor} = \frac{6}{4} = 1.5 \]

Since all three pairs give the same scale factor, we conclude that the scale factor of the dilation is 1.5.