One smaller and one larger triangle side by side.

Question

A triangle has a base of 8 cm and a height of 3 cm. The triangle is dilated by a scale factor of 3. Calculate and compare the areas of the dilated triangle and the original triangle.

(1 point)
Responses

The original triangle has an area of 12 cm2 and the dilated triangle has an area of 36 cm2; therefore, the dilated triangle has an area 3 times the size of the original triangle.
The original triangle has an area of 12 cm squared and the dilated triangle has an area of 36 cm squared ; therefore, the dilated triangle has an area 3 times the size of the original triangle.

The original triangle has an area of 12 cm2 and the dilated triangle has an area of 108 cm2; therefore, the dilated triangle has an area 9 times the size of the original triangle.
The original triangle has an area of 12 cm squared and the dilated triangle has an area of 108 cm squared ; therefore, the dilated triangle has an area 9 times the size of the original triangle.

The original triangle has an area of 24 cm2 and the dilated triangle has an area of 36 cm2; therefore, the dilated triangle has an area 1.5 times the size of the original triangle.
The original triangle has an area of 24 cm squared and the dilated triangle has an area of 36 cm squared ; therefore, the dilated triangle has an area 1.5 times the size of the original triangle.

The original triangle has an area of 24 cm2 and the dilated triangle has an area of 108 cm2; therefore, the dilated triangle has an area 4.5 times the size of the original triangle.
The original triangle has an area of 24 cm squared and the dilated triangle has an area of 108 cm squared ; therefore, the dilated triangle has an area 4.5 times the size of the original triangle.

Answer 1

To find the area of the original triangle, we can use the formula for the area of a triangle:

\[ \text{Area} = \frac{1}{2} \times \text{base} \times \text{height} \]

For the original triangle:

• Base = 8 cm
• Height = 3 cm

Calculating the area:

\[ \text{Area} = \frac{1}{2} \times 8 , \text{cm} \times 3 , \text{cm} = \frac{1}{2} \times 24 , \text{cm}^2 = 12 , \text{cm}^2 \]

Next, when the triangle is dilated by a scale factor of 3, both the base and height of the triangle increase by this scale factor.

New base:

\[ \text{New base} = 3 \times 8 , \text{cm} = 24 , \text{cm} \]

New height:

\[ \text{New height} = 3 \times 3 , \text{cm} = 9 , \text{cm} \]

Calculating the area of the dilated triangle:

\[ \text{Area} = \frac{1}{2} \times \text{new base} \times \text{new height} = \frac{1}{2} \times 24 , \text{cm} \times 9 , \text{cm} = \frac{1}{2} \times 216 , \text{cm}^2 = 108 , \text{cm}^2 \]

Now we can compare the areas. The area of the original triangle is 12 cm² and the area of the dilated triangle is 108 cm².

To find how many times larger the area of the dilated triangle is compared to the original:

\[ \frac{108 , \text{cm}^2}{12 , \text{cm}^2} = 9 \]

Thus, the dilated triangle has an area 9 times the size of the original triangle.

The correct statement is:

The original triangle has an area of 12 cm² and the dilated triangle has an area of 108 cm²; therefore, the dilated triangle has an area 9 times the size of the original triangle.

Answer 2

One smaller and one larger square side by side.
Chaz draws a square with sides equal to 4 units, then he dilates the square so the sides are equal to 6 units. Chaz writes a fraction in lowest terms to express the relationship between the area of the original square and the dilated square, with the numerator expressing the area of the dilated square. What is that fraction?

(1 point)
Responses

1636
16 over 36

49
Start Fraction 4 over 9 End Fraction

214
2 Start Fraction 1 over 4 End Fraction

23

Answer 3

To solve this problem, we first need to find the areas of both the original and the dilated squares.

The area of a square is given by the formula:

\[ \text{Area} = \text{side}^2 \]

1. Original square:

   • Side length = 4 units
   • Area = \(4^2 = 16\) square units

2. Dilated square:

   • Side length = 6 units
   • Area = \(6^2 = 36\) square units

Next, we need to express the relationship between the area of the dilated square and the area of the original square as a fraction. The fraction will have the area of the dilated square as the numerator and the area of the original square as the denominator:

\[ \text{Fraction} = \frac{\text{Area of dilated square}}{\text{Area of original square}} = \frac{36}{16} \]

Now, we simplify the fraction:

\[ \frac{36}{16} = \frac{9}{4} \]

The fraction \( \frac{9}{4} \) is in lowest terms.

Now, comparing this with the given responses, none of the options directly match \( \frac{9}{4} \).

If the task is asking for a specific outcome, please specify further to get correct options. However, from the calculations done, the correct fraction is \( \frac{9}{4} \). Please refer back to the options or verify if adjustments in the context are needed based on what is provided.

Answer 4

Kierra draws a triangle with a base of 10 cm and a height of 15 cm. She dilates the triangle using a scale factor of 45 . Then, Kierra finds the difference between the two areas by subtracting. How much greater is the area of the original triangle than the area of the dilated triangle?(1 point)
Responses

27 cm2
27 cm squared

75 cm2
75 cm squared

54 cm2
54 cm squared

48 cm2

Answer 5

To find how much greater the area of the original triangle is than the area of the dilated triangle, we first need to calculate the area of each triangle.

Step 1: Calculate the area of the original triangle

The area \( A \) of a triangle is given by the formula:

\[ A = \frac{1}{2} \times \text{base} \times \text{height} \]

For the original triangle:

• Base = 10 cm
• Height = 15 cm

Calculating the area:

\[ A_{\text{original}} = \frac{1}{2} \times 10 , \text{cm} \times 15 , \text{cm} = \frac{1}{2} \times 150 , \text{cm}^2 = 75 , \text{cm}^2 \]

Step 2: Calculate the area of the dilated triangle

Since the triangle is dilated by a scale factor of 45, we can find the new dimensions. The new base and height will be:

• New base = \( 45 \times 10 , \text{cm} = 450 , \text{cm} \)
• New height = \( 45 \times 15 , \text{cm} = 675 , \text{cm} \)

Now, we calculate the area of the dilated triangle:

\[ A_{\text{dilated}} = \frac{1}{2} \times 450 , \text{cm} \times 675 , \text{cm} \]

Calculating this area:

\[ A_{\text{dilated}} = \frac{1}{2} \times 303750 , \text{cm}^2 = 151875 , \text{cm}^2 \]

Step 3: Find the difference between the two areas

Now we find the difference between the area of the original triangle and the area of the dilated triangle:

\[ \text{Difference} = A_{\text{original}} - A_{\text{dilated}} = 75 , \text{cm}^2 - 151875 , \text{cm}^2 \]

Since \(139875 > 75\), the area of the dilated triangle is significantly larger than that of the original triangle which means the difference should be computed as:

\[ \text{Difference} = A_{\text{dilated}} - A_{\text{original}} = 151875 , \text{cm}^2 - 75 , \text{cm}^2 = 151800 , \text{cm}^2 \]

Since the question asks how much greater is the area of the original triangle than the area of the dilated triangle, the area of the original triangle is actually considerably less than that of the dilated triangle.

Conclusion

Thus, the correct response to the extent that area of the original triangle is lesser and the area of the dilated triangle is greater cannot be positive or match any provided options, given the figures involved. Would you like to clarify how to interpret the question or help in any particular aspect?

Answer 6

Determine how dilation affects the area of a pentagon when the sides of the pentagon are dilated by a scale factor less than 1.(1 point)
Responses

The area increases more rapidly than the lengths of the sides increase.
The area increases more rapidly than the lengths of the sides increase.

The area decreases more rapidly than the lengths of the sides decrease.
The area decreases more rapidly than the lengths of the sides decrease.

The area increases, but less rapidly than the lengths of the sides increase.
The area increases, but less rapidly than the lengths of the sides increase.

The area decreases, but less rapidly than the lengths of the sides decrease.

Answer 7

When a geometric figure, such as a pentagon, undergoes dilation by a scale factor less than 1, each linear dimension (such as the lengths of the sides) decreases.

To understand how this affects the area, we should consider the relationship between linear dimensions and area:

1. Linear Dimension Change: If the scale factor is \( k \) (where \( 0 < k < 1 \)), then each side of the pentagon is multiplied by \( k \).

2. Area Change: The area of a two-dimensional shape scales by the square of the scale factor. Therefore, if the sides of the pentagon are dilated by a scale factor of \( k \), the area will change by a factor of \( k^2 \).

Since \( k < 1 \):

• The area of the pentagon will decrease because when you multiply by a number less than 1, the result is smaller than the original.

Thus, we can determine the relationship:

• The area decreases more rapidly than the lengths of the sides decrease because the area is related to the square of the scale factor, while the sides only decrease by the scale factor itself.

Conclusion

The correct response is:

The area decreases more rapidly than the lengths of the sides decrease.