Question
Deion dilates this composite figure using a scale factor of 1/2. Determine the area of the dilated figure in square units.
The image shows sides 4, 3, 6
0.75 square units
3.75 square units
3 square units
15 square units
The image shows sides 4, 3, 6
0.75 square units
3.75 square units
3 square units
15 square units
Answers
GPT-4o mini
To determine the area of a dilated figure when given the original dimensions, you can use the following steps:
1. Calculate the area of the original figure.
2. Use the scale factor to find the area of the dilated figure.
Assuming the original figure is a triangle (since the sides provided (4, 3, 6) suggest it could be a triangle) we can first use Heron's formula to find the area.
**Step 1: Calculate the area of the original figure using Heron's formula.**
The sides of the triangle are \( a = 4 \), \( b = 3 \), and \( c = 6 \).
First, compute the semi-perimeter \( s \):
\[
s = \frac{a + b + c}{2} = \frac{4 + 3 + 6}{2} = \frac{13}{2} = 6.5
\]
Now, apply Heron's formula for the area \( A \):
\[
A = \sqrt{s(s-a)(s-b)(s-c)}
\]
Substituting the values:
\[
A = \sqrt{6.5 \left(6.5 - 4\right) \left(6.5 - 3\right) \left(6.5 - 6\right)}
\]
\[
= \sqrt{6.5 \cdot 2.5 \cdot 3.5 \cdot 0.5}
\]
Calculating each term:
\[
6.5 \cdot 2.5 = 16.25
\]
\[
16.25 \cdot 3.5 = 56.875
\]
\[
56.875 \cdot 0.5 = 28.4375
\]
Now take the square root:
\[
A = \sqrt{28.4375} \approx 5.34 \text{ square units}
\]
**Step 2: Determine the area of the dilated figure.**
The scale factor is \( \frac{1}{2} \). To find the area of the dilated figure, we square the scale factor and multiply by the original area:
\[
\text{Area of dilated figure} = \text{Original Area} \times \left(\frac{1}{2}\right)^2
\]
\[
= 5.34 \times \frac{1}{4} = 5.34 \times 0.25 = 1.335 \text{ square units}
\]
None of the options given (0.75, 3.75, 3, 15) are 1.335 square units. Thus, we may need to re-evaluate either the shape or given data since it does not match options.
Assuming the original shapes had an area closer to an estimate, if the original area was not calculated correctly or misidentified, check if it's simple and update accordingly. If you directly have a total area (like a rectangle or such), scaling would affect only the original area proportionately.
Final note, to guide to options check if confirming directions, if the original polygons have more than triangle areas or other shapes can lead outcomes retraced. Otherwise, re-check using base and height dimensions typically if they're supposed to be base-set shapes (rectangles or trapezoids) given earlier sides for miscounts.
Therefore, double-check the problem setup along shapes typically.
1. Calculate the area of the original figure.
2. Use the scale factor to find the area of the dilated figure.
Assuming the original figure is a triangle (since the sides provided (4, 3, 6) suggest it could be a triangle) we can first use Heron's formula to find the area.
**Step 1: Calculate the area of the original figure using Heron's formula.**
The sides of the triangle are \( a = 4 \), \( b = 3 \), and \( c = 6 \).
First, compute the semi-perimeter \( s \):
\[
s = \frac{a + b + c}{2} = \frac{4 + 3 + 6}{2} = \frac{13}{2} = 6.5
\]
Now, apply Heron's formula for the area \( A \):
\[
A = \sqrt{s(s-a)(s-b)(s-c)}
\]
Substituting the values:
\[
A = \sqrt{6.5 \left(6.5 - 4\right) \left(6.5 - 3\right) \left(6.5 - 6\right)}
\]
\[
= \sqrt{6.5 \cdot 2.5 \cdot 3.5 \cdot 0.5}
\]
Calculating each term:
\[
6.5 \cdot 2.5 = 16.25
\]
\[
16.25 \cdot 3.5 = 56.875
\]
\[
56.875 \cdot 0.5 = 28.4375
\]
Now take the square root:
\[
A = \sqrt{28.4375} \approx 5.34 \text{ square units}
\]
**Step 2: Determine the area of the dilated figure.**
The scale factor is \( \frac{1}{2} \). To find the area of the dilated figure, we square the scale factor and multiply by the original area:
\[
\text{Area of dilated figure} = \text{Original Area} \times \left(\frac{1}{2}\right)^2
\]
\[
= 5.34 \times \frac{1}{4} = 5.34 \times 0.25 = 1.335 \text{ square units}
\]
None of the options given (0.75, 3.75, 3, 15) are 1.335 square units. Thus, we may need to re-evaluate either the shape or given data since it does not match options.
Assuming the original shapes had an area closer to an estimate, if the original area was not calculated correctly or misidentified, check if it's simple and update accordingly. If you directly have a total area (like a rectangle or such), scaling would affect only the original area proportionately.
Final note, to guide to options check if confirming directions, if the original polygons have more than triangle areas or other shapes can lead outcomes retraced. Otherwise, re-check using base and height dimensions typically if they're supposed to be base-set shapes (rectangles or trapezoids) given earlier sides for miscounts.
Therefore, double-check the problem setup along shapes typically.