Question

the right triangle below is dilated by a scale factor of 5. 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.

Explanation:

Step1: Find the perimeter of the given right - triangle

Let the sides of the right - triangle be \(a = 12\), \(b = 5\). Using the Pythagorean theorem \(c=\sqrt{a^{2}+b^{2}}\), where \(c\) is the hypotenuse. So \(c=\sqrt{12^{2}+5^{2}}=\sqrt{144 + 25}=\sqrt{169}=13\). The perimeter \(P_1=a + b + c=12+5 + 13=30\) units.

Step2: Find the area of the given right - triangle

The area of a right - triangle \(A_1=\frac{1}{2}\times a\times b\). Substituting \(a = 12\) and \(b = 5\), we get \(A_1=\frac{1}{2}\times12\times5 = 30\) square units.

Step3: Find the perimeter of the dilated right - triangle

When a figure is dilated by a scale factor \(k = 5\), the perimeter of the dilated figure \(P_2=k\times P_1\). So \(P_2=5\times30 = 150\) units.

Step4: Find the area of the dilated right - triangle

When a figure is dilated by a scale factor \(k\), the area of the dilated figure \(A_2=k^{2}\times A_1\). So \(A_2=5^{2}\times30=25\times30 = 750\) square units.

Answer:

Perimeter of given right - triangle: 30 units
Perimeter of dilated right - triangle: 150 units
Area of given right - triangle: 30 square units
Area of dilated right - triangle: 750 square units