Question

a mosaic will be created inside a form with the vertices ( x(-20, -15) ), ( y(5, 5) ), and ( z(25, -20) ). what length of material in units on the scale drawing is required for the edge of the mosaic?
round the answer to the nearest tenth if necessary. enter your answer in the box.
(square) units

Explanation:

Step1: Find the length of XY

The distance formula between two points \((x_1,y_1)\) and \((x_2,y_2)\) is \(d = \sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}\). For \(X(-20,-15)\) and \(Y(5,5)\), we have \(x_1=-20,y_1 = - 15,x_2 = 5,y_2=5\).
\[

$$\begin{align*} XY&=\sqrt{(5 - (-20))^2+(5 - (-15))^2}\\ &=\sqrt{(25)^2+(20)^2}\\ &=\sqrt{625 + 400}\\ &=\sqrt{1025}\\ &\approx32.0 \end{align*}$$

\]

Step2: Find the length of YZ

For \(Y(5,5)\) and \(Z(25,-20)\), \(x_1 = 5,y_1=5,x_2=25,y_2=-20\).
\[

$$\begin{align*} YZ&=\sqrt{(25 - 5)^2+(-20 - 5)^2}\\ &=\sqrt{(20)^2+(-25)^2}\\ &=\sqrt{400+625}\\ &=\sqrt{1025}\\ &\approx32.0 \end{align*}$$

\]

Step3: Find the length of ZX

For \(Z(25,-20)\) and \(X(-20,-15)\), \(x_1 = 25,y_1=-20,x_2=-20,y_2=-15\).
\[

$$\begin{align*} ZX&=\sqrt{(-20 - 25)^2+(-15 - (-20))^2}\\ &=\sqrt{(-45)^2+(5)^2}\\ &=\sqrt{2025 + 25}\\ &=\sqrt{2050}\\ &\approx45.3 \end{align*}$$

\]

Step4: Find the perimeter

The perimeter \(P=XY + YZ+ZX\). Substituting the values: \(P\approx32.0+32.0 + 45.3=109.3\)

Answer:

\(109.3\)