Question

The side lengths of triangle ABC are 10, 14 and 16. The side lengths of triangle A'B'C' Are 35, 49, and 56. Identify the scale factor of a single dialation between the pre image ABC and image A'B'C'

Answer 1

[AI-generated answer — Bot GPT 3.5]

The scale factor of a single dilation between two similar figures is given by the ratio of the corresponding side lengths.

In this case, the side lengths of triangle ABC are 10, 14, and 16, while the side lengths of triangle A'B'C' are 35, 49, and 56.

The scale factor can be found by dividing the corresponding side lengths of A'B'C' by the corresponding side lengths of ABC:

35/10 = 3.5
49/14 = 3.5
56/16 = 3.5

The scale factor is 3.5 for each side length.

Answer 2

Describe the dialation of triangle , a b c with a scale factor of five and a centerpoint of dilation at the origin 0,0

Answer 3

[AI-generated answer — Bot GPT 3.5]

When a triangle ABC is dilated, the resulting image, let's call it A'B'C', is created by multiplying the coordinates of each vertex of triangle ABC by the scale factor.

In this case, the scale factor is 5, and the center point of dilation is at the origin (0,0).

Let's assume the coordinates of the vertices of triangle ABC are:

A(x1, y1)
B(x2, y2)
C(x3, y3)

To find the coordinates of the vertices of A'B'C', we can multiply the original coordinates by the scale factor:

A'(5x1, 5y1)
B'(5x2, 5y2)
C'(5x3, 5y3)

Therefore, the dilation of triangle ABC with a scale factor of 5 and a center point of dilation at the origin (0,0) results in triangle A'B'C' with vertices A'(5x1, 5y1), B'(5x2, 5y2), and C'(5x3, 5y3).

Answer 4

Is this five times larger or smaller and is this five times farther or closer

Answer 5

[AI-generated answer — Bot GPT 3.5]

When a triangle is dilated with a scale factor greater than 1, like in this case where the scale factor is 5, the resulting image triangle will be larger.

Since each coordinate of the vertices is multiplied by 5, the resulting triangle A'B'C' will be 5 times larger in each dimension (length, width, and height) compared to the original triangle ABC.

As for the distance, the scale factor of 5 indicates that each point of the resulting triangle A'B'C' is 5 times farther away from the origin (0,0) compared to the corresponding point in the original triangle ABC.

Answer 6

Triangle ABC has vertices point A -2, 4, and pointB -6, 8 and point C -,2, 4. Is after dilation, triangle A'B'C' has vertices point A' -1, 2, POINT B' -3, 4, AND POINT C' -1, 2. The turn the scale factor for the dialation of triangle ABC With a center point of dialation at the origin 0,0

Answer 7

[AI-generated answer — Bot GPT 3.5]

To determine the scale factor for the dilation of triangle ABC, we compare the corresponding side lengths of the original and dilated triangles.

Let's calculate the side lengths of triangle ABC:

Side AB:
Distance = √((-6 - (-2))^2 + (8 - 4)^2) = √((-4)^2 + 4^2) = √(16 + 16) = √32

Side BC:
Distance = √(((-6) - (-2))^2 + (8 - (-2))^2) = √((-4)^2 + 10^2) = √(16 + 100) = √116

Side AC:
Distance = √((-6 - (-2))^2 + ((-2) - 4)^2) = √((-4)^2 + (-6)^2) = √(16 + 36) = √52

Now let's calculate the corresponding side lengths of triangle A'B'C':

Side A'B':
Distance = √((-3 - (-1))^2 + (4 - 2)^2) = √((-2)^2 + 2^2) = √(4 + 4) = √8

Side B'C':
Distance = √((-3 - (-1))^2 + (4 - 2)^2) = √((-2)^2 + 2^2) = √(4 + 4) = √8

Side A'C':
Distance = √((-1 - (-1))^2 + (2 - 2)^2) = √(0^2 + 0^2) = √0 = 0

Comparing the side lengths, we see that the side AB of the dilated triangle is √8, which is approximately 2.82, while the original side AB is √32, which is approximately 5.66.

Therefore, the scale factor for the dilation of triangle ABC is approximately 2.82/5.66, or approximately 0.5.

Answer 8

No

Answers are -2 , 2 ,-1/2 , and 1/2

Answer 9

[AI-generated answer — Bot GPT 3.5]

Apologies for the error. Let's calculate the scale factor again.

The coordinates of the original triangle ABC are:

A: (-2, 4)
B: (-6, 8)
C: (-2, 4)

The coordinates of the dilated triangle A'B'C' are:

A': (-1, 2)
B': (-3, 4)
C': (-1, 2)

To find the scale factor, we can compare the corresponding side lengths of the original and dilated triangles.

Side AB:
Distance(ABC) = √((-6 - (-2))^2 + (8 - 4)^2) = √((-4)^2 + 4^2) = √(16 + 16) = √32

Distance(A'B'C') = √((-3 - (-1))^2 + (4 - 2)^2) = √((-2)^2 + 2^2) = √(4 + 4) = √8

Scale factor for side AB = Distance(A'B'C') / Distance(ABC) = √8 / √32 = (√8 / √32) * (√32 / √32) = √(8/32) = √(1/4) = 1/2

Similarly, we can calculate the scale factor for side AC or side BC and find that they are also 1/2.

Therefore, the scale factor for the dilation of triangle ABC is 1/2.