Question

the vertices of rhombus defg are d(1, 4), e(4, 0), f(1, -4), and g(-2, 0). what is the perimeter of the rhombus? \boxed{} units

Explanation:

Step1: Recall rhombus perimeter formula

A rhombus has 4 equal sides, so perimeter \( P = 4s \), where \( s \) is the length of one side. We can find the length of a side using the distance formula between two vertices. Let's find the distance between \( D(1, 4) \) and \( E(4, 0) \). The distance formula is \( d=\sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2} \).

Step2: Apply distance formula to D and E

For \( D(1, 4) \) and \( E(4, 0) \), \( x_1 = 1, y_1 = 4, x_2 = 4, y_2 = 0 \). Plug into the formula:
\( d=\sqrt{(4 - 1)^2+(0 - 4)^2}=\sqrt{3^2+(-4)^2}=\sqrt{9 + 16}=\sqrt{25}=5 \).
So one side length \( s = 5 \).

Step3: Calculate perimeter

Using \( P = 4s \), substitute \( s = 5 \):
\( P = 4\times5 = 20 \).

Answer:

20