Question

Line segment AB
has endpoints A (−6, 4)
and B (−8, 2)
. Find the coordinates of A′
and B′
after a dilation with a scale factor of 1/2
, centered at the origin.(1 point)

A coordinate plane with four quadrants shows x-axis ranging from negative 5 to 12 in increments of 1 and the y-axis from 17 to negative 4 in increments of 1. Points upper A, upper B, upper A prime, and upper B prime are plotted on quadrant 1. The plotted points are as follows: Upper A is left parenthesis 2 comma 3 right parenthesis and upper B is left parenthesis 4 comma 1 right parenthesis. The points are joined by a line. Upper A prime is plotted as left parenthesis 6 comma 9 right parenthesis and upper B prime as left parenthesis 12 comma 3 right parenthesis. The points are joined by a line.
Determine the scale factor used in the dilation of line segment AB
, centered at the origin.

(1 point)
The scale factor is
.


Use the image to answer the question.

A coordinate plane with four quadrants shows an x axis capped with arrows and ranging from negative 11 to 10 in 1 unit increments and a y axis capped with arrows and ranging from 12 to negative 12 in 1 unit increments. Points upper A, upper B, upper A prime, and upper B prime are plotted. A coordinate plane with four quadrants shows an x axis capped with arrows and ranging from negative 11 to 10 in 1 unit increments and a y axis capped with arrows and ranging from 12 to negative 12 in 1 unit increments. Point upper A is plotted as left parenthesis negative 3 comma 4 right parenthesis and point upper B is plotted as left parenthesis 2 comma negative 4 right parenthesis. The points are joined by a line that extends beyond the points and is capped with arrows at both ends. Point upper A prime is plotted as left parenthesis negative 6 comma 8 right parenthesis and point upper B prime is plotted as left parenthesis 4 comma negative 8 right parenthesis. The points are joined by a line that extends beyond the points and is capped with arrows at both ends.

Determine the scale factor used in the dilation of line AB
, with a center point of dilation at the origin.

(1 point)
The scale factor is
.




Triangle ABC
has vertices point A(2,1)
, 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 7 and a center point of dilation at the origin (0,0)
.(2 points)


Triangle ABC
has vertices point A(2, 1)
, point B(−2, 2)
, and point C(−3,−1)
. After dilation, triangle A′B′C′
has vertices point A′(4, 2)
, point B′(−4, 4)
, and point C′(−6,−2)
. Determine the scale factor for the dilation of triangle ABC
with a center point of dilation at the origin (0, 0 )
.(1 point)

Answers

GPT-4o mini
To find the coordinates after a dilation, we can use the formula for dilation from the origin. The formula for a point \((x, y)\) after dilation with a scale factor \(k\) is given by:

\[
(x', y') = (kx, ky)
\]

### 1. Dilation of Segment AB

**Endpoints:**
- A \((-6, 4)\)
- B \((-8, 2)\)

**Scale factor:** \( \frac{1}{2} \)

**Coordinates of A' and B':**
For point A:
\[
A' = \left(\frac{1}{2} \cdot (-6), \frac{1}{2} \cdot 4\right) = (-3, 2)
\]

For point B:
\[
B' = \left(\frac{1}{2} \cdot (-8), \frac{1}{2} \cdot 2\right) = (-4, 1)
\]

Thus, the coordinates of \(A'\) and \(B'\) after the dilation are:
**A' = (-3, 2)**
**B' = (-4, 1)**

---

### 2. Determine the Scale Factor Used in the Dilation of Segment AB

From the provided points in the image:
- A = (2, 3) and B = (4, 1)
- A' = (6, 9) and B' = (12, 3)

**To find the scale factor:**

For A:
\[
\text{Scale factor for A} = \frac{\text{A'}_x}{\text{A}_x} = \frac{6}{2} = 3
\]

For B:
\[
\text{Scale factor for B} = \frac{\text{B'}_x}{\text{B}_x} = \frac{12}{4} = 3
\]

Thus, the scale factor for this dilation is \( \text{Scale factor} = 3\).

---

### 3. Dilation of Triangle ABC

**Vertices:**
- A(2, 1)
- B(4, 3)
- C(−2, 5)

**Scale factor:** \(7\)

**Coordinates of A', B', and C':**
For point A:
\[
A' = (7 \cdot 2, 7 \cdot 1) = (14, 7)
\]

For point B:
\[
B' = (7 \cdot 4, 7 \cdot 3) = (28, 21)
\]

For point C:
\[
C' = (7 \cdot -2, 7 \cdot 5) = (-14, 35)
\]

Thus, the coordinates after dilation are:
**A' = (14, 7)**
**B' = (28, 21)**
**C' = (-14, 35)**

---

### 4. Determine the Scale Factor for Dilation of Triangle ABC

**Vertices:**
- A(2, 1)
- B(−2, 2)
- C(−3,−1)

**After dilation:**
- A'(4, 2)
- B'(-4, 4)
- C'(-6, -2)

**To find the scale factor:**

For A:
\[
\text{Scale factor for A} = \frac{\text{A'}_x}{\text{A}_x} = \frac{4}{2} = 2
\]

For B:
\[
\text{Scale factor for B} = \frac{\text{B'}_x}{\text{B}_x} = \frac{-4}{-2} = 2
\]

For C:
\[
\text{Scale factor for C} = \frac{\text{C'}_y}{\text{C}_y} = \frac{-2}{-1} = 2
\]

Thus, the scale factor for this dilation is \(2\).

---

### Summary:
- **Dilation of segment AB:**
A' = (-3, 2)
B' = (-4, 1)
- **Scale factor for segment AB:** \( 3 \)
- **Dilation of triangle ABC:**
A' = (14, 7)
B' = (28, 21)
C' = (-14, 35)
- **Scale factor for triangle ABC:** \( 2 \)

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