Math / Geometry

Question5. If $A B$ is 12 , what is the length of $A^{\prime} B^{\prime}$ ? (Lesson 3-5)
6. Right angle $A B C$ is taken by a dilation with center $P$ and scale factor $\frac{1}{2}$ to angle $A^{\prime} B^{\prime} C^{\prime}$ . What is the measure of angle $A^{\prime} B^{\prime} C^{\prime}$ ? (Lesson 3-4)
7. Respond to each question. (Lesson 3-4)

Studdy Solution

STEP 1

1. Dilation is a transformation that produces an image that is the same shape as the original, but is a different size.
2. The scale factor determines how much larger or smaller the image will be compared to the original.
3. Angles remain unchanged under dilation.

STEP 2

1. Calculate the length of $A^{\prime} B^{\prime}$ .
2. Determine the measure of angle $A^{\prime} B^{\prime} C^{\prime}$ .
3. Respond to each question.

STEP 3

Given that $AB = 12$ and the scale factor of the dilation is $\frac{1}{2}$ , the length of $A^{\prime} B^{\prime}$ can be calculated by multiplying the original length by the scale factor:
$A^{\prime} B^{\prime} = AB \times \text{scale factor}$ $A^{\prime} B^{\prime} = 12 \times \frac{1}{2}$

STEP 4

Perform the multiplication:
$A^{\prime} B^{\prime} = 12 \times \frac{1}{2} = 6$

STEP 5

Since dilation does not affect angles, the measure of angle $A^{\prime} B^{\prime} C^{\prime}$ is the same as the measure of angle $ABC$ . Given that $ABC$ is a right angle, we have:
$\text{Measure of } \angle A^{\prime} B^{\prime} C^{\prime} = 90^\circ$

STEP 6

Respond to each question: - The length of $A^{\prime} B^{\prime}$ is $6$ . - The measure of angle $A^{\prime} B^{\prime} C^{\prime}$ is $90^\circ$ .
The solutions are: - Length of $A^{\prime} B^{\prime}$ : $\boxed{6}$ - Measure of angle $A^{\prime} B^{\prime} C^{\prime}$ : $\boxed{90^\circ}$

Was this helpful?

Studdy Solution

STEP 1

1. Dilation is a transformation that produces an image that is the same shape as the original, but is a different size.
2. The scale factor determines how much larger or smaller the image will be compared to the original.
3. Angles remain unchanged under dilation.

STEP 2

1. Calculate the length of $A^{\prime} B^{\prime}$ .
2. Determine the measure of angle $A^{\prime} B^{\prime} C^{\prime}$ .
3. Respond to each question.

STEP 3

Given that $AB = 12$ and the scale factor of the dilation is $\frac{1}{2}$ , the length of $A^{\prime} B^{\prime}$ can be calculated by multiplying the original length by the scale factor:
$A^{\prime} B^{\prime} = AB \times \text{scale factor}$ $A^{\prime} B^{\prime} = 12 \times \frac{1}{2}$

STEP 4

Perform the multiplication:
$A^{\prime} B^{\prime} = 12 \times \frac{1}{2} = 6$

STEP 5

Since dilation does not affect angles, the measure of angle $A^{\prime} B^{\prime} C^{\prime}$ is the same as the measure of angle $ABC$ . Given that $ABC$ is a right angle, we have:
$\text{Measure of } \angle A^{\prime} B^{\prime} C^{\prime} = 90^\circ$

STEP 6

Respond to each question: - The length of $A^{\prime} B^{\prime}$ is $6$ . - The measure of angle $A^{\prime} B^{\prime} C^{\prime}$ is $90^\circ$ .
The solutions are: - Length of $A^{\prime} B^{\prime}$ : $\boxed{6}$ - Measure of angle $A^{\prime} B^{\prime} C^{\prime}$ : $\boxed{90^\circ}$

Get Studdy

Studdy solves anything!

Start learning now

Download Studdy AI Tutor now. Learn with ease and get all help you need to be successful at school.

Studdy Solution

STEP 1

1. Dilation is a transformation that produces an image that is the same shape as the original, but is a different size.2. The scale factor determines how much larger or smaller the image will be compared to the original.3. Angles remain unchanged under dilation.

STEP 2

1. Calculate the length of A′B′ A^{\prime} B^{\prime} A′B′.2. Determine the measure of angle A′B′C′ A^{\prime} B^{\prime} C^{\prime} A′B′C′.3. Respond to each question.

STEP 3

STEP 4

Perform the multiplication:A′B′=12×12=6 A^{\prime} B^{\prime} = 12 \times \frac{1}{2} = 6 A′B′=12×21​=6

STEP 5

STEP 6

Get Studdy

Studdy solves anything!

Start learning now

Download Studdy AI Tutor now. Learn with ease and get all help you need to be successful at school.

1. Dilation is a transformation that produces an image that is the same shape as the original, but is a different size.
2. The scale factor determines how much larger or smaller the image will be compared to the original.
3. Angles remain unchanged under dilation.

1. Calculate the length of $A^{\prime} B^{\prime}$ .
2. Determine the measure of angle $A^{\prime} B^{\prime} C^{\prime}$ .
3. Respond to each question.

Perform the multiplication:
$A^{\prime} B^{\prime} = 12 \times \frac{1}{2} = 6$