Question

question
the right triangle below is dilated by a scale factor of 3. find the perimeter and area of the right triangle below, as well as the perimeter and area of the dilated right triangle. express your answers as whole numbers, decimals, or fractions in simplest terms (no mixed numbers).
figures are not necessarily drawn to scale.
answer attempt 1 out of 2
perimeter of given right triangle units
area of given right triangle units²
perimeter of dilated right triangle units
area of dilated right triangle units²

Explanation:

Step1: Calculate the perimeter of the original triangle

The perimeter \(P\) of a triangle is the sum of its side - lengths. For the original right - triangle with sides \(a = 8\), \(b=15\), and \(c = 17\), \(P_{original}=8 + 15+17\).

$$P_{original}=40$$

Step2: Calculate the area of the original triangle

The area \(A\) of a right - triangle is given by \(A=\frac{1}{2}\times base\times height\). Here, base \(b = 8\) and height \(h = 15\).

$$A_{original}=\frac{1}{2}\times8\times15=60$$

Step3: Use the properties of dilation

If a figure is dilated by a scale factor \(k\), the perimeter of the dilated figure \(P_{dilated}\) is related to the perimeter of the original figure \(P_{original}\) by the formula \(P_{dilated}=k\times P_{original}\). Given \(k = 3\), \(P_{dilated}=3\times40 = 120\).
If a figure is dilated by a scale factor \(k\), the area of the dilated figure \(A_{dilated}\) is related to the area of the original figure \(A_{original}\) by the formula \(A_{dilated}=k^{2}\times A_{original}\). Given \(k = 3\), \(A_{dilated}=3^{2}\times60=9\times60 = 540\)

Answer:

Perimeter of given right - triangle: \(40\) units
Area of given right - triangle: \(60\) units²
Perimeter of dilated right - triangle: \(120\) units
Area of dilated right - triangle: \(540\) units²