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

Question

. 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

75 cm2
75 cm squared

54 cm2
54 cm squared

48 cm2
48 cm squared

27 cm2

Answer 1

To solve the problem, we first need to find the area of the original triangle and then the area of the dilated triangle.

1. Area of the original triangle: \[ \text{Area} = \frac{1}{2} \times \text{base} \times \text{height} \] Given:

   • Base = 10 cm
   • Height = 15 cm

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

2. Dimensions of the dilated triangle: The dilation scale factor is 45, so we multiply both the base and the height by 45.

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

3. Area of the dilated triangle: \[ \text{Area} = \frac{1}{2} \times \text{new base} \times \text{new height} \] \[ \text{Area} = \frac{1}{2} \times 450 \times 675 \]

   To compute this: \[ \text{Area} = \frac{1}{2} \times 303750 = 151875 , \text{cm}^2 \]

4. Difference in areas: Now we find the difference between the areas of the original and dilated triangles: \[ \text{Difference} = \text{Area of original triangle} - \text{Area of dilated triangle} \]

   Since the dilated triangle has a much larger area compared to the original, we'll find the difference: \[ \text{Difference} = 75 , \text{cm}^2 - 151875 , \text{cm}^2 = -151800 , \text{cm}^2 \]

However, the question asks how much greater the original triangle is than the dilated triangle. The result illustrates that the original triangle is not greater at all; rather, it is significantly smaller. However, if we're just looking for the area difference magnitude, we find the area of the original triangle to be far less than the dilated triangle, confirming the negation.

The original triangle is 75 cm\(^2\) and the dilated one is 151875 cm\(^2\) making the area of the dilated triangle much larger by 151800 cm\(^2\).

Thereby, the confusion in assessments would boil down to looking at the positive magnitude of area differences.

Finally, to conclude: the original triangle is not greater than the dilated triangle; the area of the dilated triangle is greater than the original triangle. The area difference is simply framed incorrectly in terms of needing an absolute value.

So, the options provided do not apply; the area of the original triangle is much less than that of the dilated triangle.