Question

1. Determine the scale factor used in the dilation of line segment AB , centered at the origin. A (1, -3) B (2, -4) A' (4, -12) B (8, -16)

2. Determine the scale factor used in the dilation of line segment AB , with a center point of dilation at the origin. A (9, 9) B (6, 6) A' (3, 3) B' (2, 2)

3. Triangle ABC has vertices point A(−2,−2) , point B(−4,−3) , and point C(−2,−5) . Find the coordinates of A′ , B′ , and C′ after a dilation with a scale factor of 3 and a center point of dilation at the origin.

4. Dana draws an F on a clear piece of plastic. Which transformation would fail to preserve congruence?

5. If you would like to dilate a square with a side length of 8 units to create a square with a side length of 20 units, what scale factor should you use? Write your answer in decimal form.

6. Which equation shows the relation between the vertices of a dilated figure and the vertices of the original figure when the original figure is dilated using a scale factor of 5?

7. △CDE∼△PQR . m∠C=13° , m∠D=65° . What are the measures of angles P , Q , and R ?

8. Given △ABC∼△DEF , m∠A=52° , and m∠E=11° , what is the measure of angle C ?

9. Given △PQR∼△LMN , m∠Q=113° , and m∠R=26° , what is the measure of angle N ?

Answer 1

1. To find the scale factor in the dilation of line segment AB, we can use the coordinates of the points:

Points A and B are given as:
- A(1, -3)
- B(2, -4)

And points A' and B' after dilation are:
- A'(4, -12)
- B'(8, -16)

The scale factor \( k \) can be calculated using the formula:
\[
k = \frac{\text{new coordinate}}{\text{original coordinate}}
\]

Calculating for point A:
\[
k = \frac{4}{1} = 4 \quad \text{and} \quad k = \frac{-12}{-3} = 4
\]

Calculating for point B:
\[
k = \frac{8}{2} = 4 \quad \text{and} \quad k = \frac{-16}{-4} = 4
\]

Thus, the scale factor used in the dilation of line segment AB is **4**.

2. For the second dilation:
Points A and B are:
- A(9, 9)
- B(6, 6)

And their dilated points are:
- A'(3, 3)
- B'(2, 2)

Calculating the scale factor \( k \) for A:
\[
k = \frac{3}{9} = \frac{1}{3}
\]

Calculating for B:
\[
k = \frac{2}{6} = \frac{1}{3}
\]

The scale factor used in the dilation of line segment AB is **\(\frac{1}{3}\)**.

3. For triangle ABC after dilation:
Original vertices:
- A(−2,−2)
- B(−4,−3)
- C(−2,−5)

Using a scale factor of 3, we calculate the coordinates of A', B', and C':
- A'(−2 * 3, −2 * 3) = (−6, −6)
- B'(−4 * 3, −3 * 3) = (−12, −9)
- C'(−2 * 3, −5 * 3) = (−6, −15)

So, the coordinates after dilation are:
- A' = (−6, −6),
- B' = (−12, −9),
- C' = (−6, −15).

4. The transformation that would fail to preserve congruence is **dilation** because it changes the size of the figure, while translations, rotations, and reflections preserve congruence.

5. To dilate a square from a side length of 8 units to a side length of 20 units, the scale factor \( k \) can be calculated as follows:
\[
k = \frac{\text{new side length}}{\text{original side length}} = \frac{20}{8} = 2.5
\]
So, the scale factor is **2.5**.

6. For a dilation with a scale factor of \( k \), the relation between the vertices of the original figure and the dilated figure can be given by:
\[
(x', y') = (k \cdot x, k \cdot y)
\]
For a scale factor of 5, it is:
\[
(x', y') = (5x, 5y)
\]

7. Since △CDE∼△PQR, corresponding angles are equal. Therefore:
- m∠P = m∠C = 13°
- m∠Q = m∠D = 65°
To find m∠R, we can use the fact that the sum of angles in a triangle is 180°:
\[
m∠R = 180° - m∠C - m∠D = 180° - 13° - 65° = 102°.
\]

Thus the measures of angles P, Q, and R are:
- m∠P = 13°
- m∠Q = 65°
- m∠R = 102°.

8. Given △ABC∼△DEF , we have m∠E = 11°. Since corresponding angles are equal:
\[
m∠C = m∠A = 52°.
\]
So, the measure of angle C is **52°**.

9. For △PQR∼△LMN, we have:
- m∠Q = 113°,
- m∠R = 26°.

We find m∠N as follows:
\[
m∠N = 180° - m∠Q - m∠R = 180° - 113° - 26° = 41°.
\]

So, the measure of angle N is **41°**.