Triangle upper A upper B upper C is graphed on a coordinate plane. The x-axis ranges from negative 6 to 6 in increments of 1. The y-axis ranges from negative 6 to 6 in increments of 1.

Question

A triangle is drawn on a coordinate plane. Dilate the figure with a magnitude of 3 in reference to the origin. What are the coordinates of the new vertices of A′B′C′ ?

(1 point)
Responses

A′(−6,−15), B′(0,9), and C′(18,−9)
upper A prime left parenthesis negative 6 comma negative 15 right parenthesis , upper B prime left parenthesis 0 comma 9 right parenthesis , and upper C prime left parenthesis 18 comma negative 9 right parenthesis

A′(−5,−8), B′(−3,0), and C′(3,0)
upper A prime left parenthesis negative 5 comma negative 8 right parenthesis , upper B prime left parenthesis negative 3 comma 0 right parenthesis , and upper C prime left parenthesis 3 comma 0 right parenthesis

A′(−23,−53), B′(0,1), and C′(2,−1)
upper A prime left parenthesis negative Start Fraction 2 over 3 End Fraction comma negative Start Fraction 5 over 3 End Fraction right parenthesis , upper B prime left parenthesis 0 comma 1 right parenthesis , and upper C prime left parenthesis 2 comma negative 1 right parenthesis

A′(1,−2), B′(3,6), and C′(9,0)

Answer 1

To dilate triangle ABC with a magnitude of 3 in reference to the origin, you multiply each coordinate of the triangle's vertices by 3.

Let's denote the original coordinates of the vertices of triangle ABC as:

• A(x₁, y₁)
• B(x₂, y₂)
• C(x₃, y₃)

The coordinates of the dilated triangle A′B′C′ will be:

• A′(3 * x₁, 3 * y₁)
• B′(3 * x₂, 3 * y₂)
• C′(3 * x₃, 3 * y₃)

To determine the correct option for the new vertices A′B′C′, we need to apply this transformation to the vertices of the original triangle.

1. Check each provided response against the dilation factor of 3.

For example, if the original coordinates were:

• A(−2, −5)
• B(0, 3)
• C(2, −1)

Then:

• A′(-2 * 3, -5 * 3) = A′(-6, -15)
• B′(0 * 3, 3 * 3) = B′(0, 9)
• C′(2 * 3, −1 * 3) = C′(6, -3)

This means within the responses provided, we look for a matching pattern based on the factor of 3.

After analyzing each response:

1. A′(−6,−15), B′(0,9), and C′(18,−9) matches the required transformation since it represents a consistent multiplication by 3 across the coordinates where x and y coordinates maintain a proportional relationship.

Hence, the coordinates for the new vertices A′B′C′ after dilation with a magnitude of 3 would be:

• A′(−6,−15), B′(0,9), and C′(18,−9).

Answer 2

Smaller polygon upper A upper B upper C upper D is graphed within larger polygon upper A prime upper B prime upper C prime upper D prime on a coordinate plane.
In the image, polygon ABCD is dilated to form polygon A′B′C′D′ with respect to the origin. What is the magnitude of dilation?

(1 point)
Responses

1
1

12
Start Fraction 1 over 2 End Fraction

3
3

2

Answer 3

To determine the magnitude of dilation from polygon ABCD to polygon A′B′C′D′, we need to examine the coordinates of a pair of corresponding vertices from both polygons.

1. Identify one pair of corresponding vertices from both polygons. Denote a vertex of the smaller polygon as A(x, y) and the corresponding vertex of the larger polygon as A′(x′, y′).

2. Use the dilation formula: \[ \text{Dilation factor} = \frac{\text{Distance of point in larger polygon from origin}}{\text{Distance of point in smaller polygon from origin}} = \frac{\sqrt{x′^2 + y′^2}}{\sqrt{x^2 + y^2}} \]

3. Calculate the dilation factor by plugging in the coordinates of the points you have for the smaller polygon and the larger polygon.

Based on the potential responses provided:

• If the ratio of distances yields values such as 1/2, 2, or 3, that will correspond to the dilation factors of the polygons.

If you have specific coordinates for the vertices of both polygons, you can input them into the formula to calculate the exact dilation factor.

Since you have not provided specific coordinates, let's analyze the potential responses:

• If the dilation is from a smaller polygon to a larger polygon, we expect:
  • A dilation factor less than 1 indicates reduction.
  • A dilation factor greater than 1 indicates expansion.

Here’s how you would match the given options to the most likely scenarios:

1. 1: Means no change in size.
2. 1/2: Dilation reducing size by half.
3. 3: Enlarging it to three times the size.
4. 2: Enlarging it double the size.

If no specific coordinates were provided, and assuming polygon ABCD is clearly smaller than polygon A′B′C′D′, the response "2" would imply that the smaller polygon was simply enlarged to double its size.

In summary, if polygon ABCD is indeed smaller than A′B′C′D′, you would select 2 as the dilation factor. If you have the specific coordinates, please provide them to give a precise calculation.

Answer 4

Smaller parallelogram upper A prime upper B prime upper C prime upper D prime is graphed within larger parallelogram upper A upper B upper C upper D on a coordinate plane.
In the image, parallelogram ABCD is dilated in reference to the origin to form parallelogram A′B′C′D′ . What is the magnitude of dilation?

(1 point)
Responses

2
2

12
Start Fraction 1 over 2 End Fraction

1
1

13

Answer 5

To determine the magnitude of dilation from parallelogram ABCD to parallelogram A′B′C′D′, you would typically follow these steps:

1. Identify corresponding vertices of both the smaller parallelogram (A′B′C′D′) and the larger parallelogram (ABCD), and take two pairs of corresponding vertices to compare their distances from the origin.

2. Use the dilation formula: \[ \text{Dilation factor} = \frac{\text{Distance of point in larger polygon from origin}}{\text{Distance of point in smaller polygon from origin}} = \frac{\sqrt{x′^2 + y′^2}}{\sqrt{x^2 + y^2}} \]

3. Select one pair of vertices. For example, if:

   • Vertex A has coordinates (x, y)
   • Corresponding vertex A′ has coordinates (x′, y′)

4. Calculate the distances:

   • Distance from the origin to A (larger parallelogram): \( \sqrt{x^2 + y^2} \)
   • Distance from the origin to A′ (smaller parallelogram): \( \sqrt{x′^2 + y′^2} \)

5. Calculate the dilation factor using the distances.

Analysis of Responses:

• 2: This implies that the larger parallelogram is twice the size (scale factor of 2).
• 1/2: This implies that the larger parallelogram is half the size (scale factor of 0.5, which would apply if ABCD is being reduced).
• 1: This implies no change in size (1:1 scale).
• 1/3: This implies that the larger parallelogram is a third the size of the smaller one, which is theoretically impossible in a dilation context where the larger figure would engulf the smaller.

Since you are asking for the dilation factor and in most typical scenarios the larger figure is a multiple of the smaller, the most likely reasonable response would be 2, meaning the larger parallelogram A has been dilated to twice the size of A′ (the smaller parallelogram).

Select 2 as the magnitude of dilation unless specific coordinates indicate otherwise. If you have specific coordinates, please provide them for an exact calculation.