Question

the right triangle below is dilated by a scale factor of 1/3. 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 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 perimeter of given triangle

Add side - lengths: $30 + 72+78 = 180$

Step2: Calculate area of given triangle

Use area formula for right - triangle $A=\frac{1}{2}\times base\times height$. Here, base = 72 and height = 30, so $A=\frac{1}{2}\times72\times30= 1080$

Step3: Calculate side - lengths of dilated triangle

Multiply each side - length of the original triangle by the scale factor $\frac{1}{3}$. New side - lengths are $30\times\frac{1}{3}=10$, $72\times\frac{1}{3}=24$, $78\times\frac{1}{3}=26$

Step4: Calculate perimeter of dilated triangle

Add new side - lengths: $10 + 24+26 = 60$

Step5: Calculate area of dilated triangle

Use area formula with new base = 24 and new height = 10. $A=\frac{1}{2}\times24\times10 = 120$

Answer:

Perimeter of given right triangle: 180 units
Area of given right triangle: 1080 units²
Perimeter of dilated right triangle: 60 units
Area of dilated right triangle: 120 units²