Question

the right triangle below is dilated by a scale factor of 1/2. 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: Calculate perimeter of original triangle

The perimeter $P$ of a triangle is the sum of its side - lengths. For the original right - triangle with side - lengths $a = 48$, $b = 20$, and $c = 52$, $P_{original}=48 + 20+52=120$ units.

Step2: Calculate area of original triangle

The area $A$ of a right - triangle is given by $A=\frac{1}{2}\times base\times height$. Using base $b = 48$ and height $h = 20$, $A_{original}=\frac{1}{2}\times48\times20 = 480$ square units.

Step3: Calculate perimeter of dilated triangle

When 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 $P_{dilated}=k\times P_{original}$. Here, $k=\frac{1}{2}$, so $P_{dilated}=\frac{1}{2}\times120 = 60$ units.

Step4: Calculate area of dilated triangle

When 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 $A_{dilated}=k^{2}\times A_{original}$. Here, $k = \frac{1}{2}$, so $A_{dilated}=(\frac{1}{2})^{2}\times480=\frac{1}{4}\times480 = 120$ square units.

Answer:

Perimeter of given right triangle: 120 units
Area of given right triangle: 480 units²
Perimeter of dilated right triangle: 60 units
Area of dilated right triangle: 120 units²