Question

1. find the perimeter of the triangle abd:

image of a coordinate grid with points a(1, 4), b(-3, 1), d(4, 0), c(0, -3) plotted
a. (5sqrt{2} + 10)
b. (5sqrt{2})
c. 10
d. (10sqrt{2})

Explanation:

Step1: Recall the distance formula

The distance between two points \((x_1,y_1)\) and \((x_2,y_2)\) is \(d = \sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}\). We need to find the lengths of \(AB\), \(BD\), and \(AD\) to calculate the perimeter of \(\triangle ABD\).

Step2: Calculate the length of \(AB\)

For points \(A(1,4)\) and \(B(-3,1)\):
\[

$$\begin{align*} AB&=\sqrt{(-3 - 1)^2+(1 - 4)^2}\\ &=\sqrt{(-4)^2+(-3)^2}\\ &=\sqrt{16 + 9}\\ &=\sqrt{25}\\ & = 5 \end{align*}$$

\]

Step3: Calculate the length of \(AD\)

For points \(A(1,4)\) and \(D(4,0)\):
\[

$$\begin{align*} AD&=\sqrt{(4 - 1)^2+(0 - 4)^2}\\ &=\sqrt{(3)^2+(-4)^2}\\ &=\sqrt{9 + 16}\\ &=\sqrt{25}\\ & = 5 \end{align*}$$

\]

Step4: Calculate the length of \(BD\)

For points \(B(-3,1)\) and \(D(4,0)\):
\[

$$\begin{align*} BD&=\sqrt{(4 - (-3))^2+(0 - 1)^2}\\ &=\sqrt{(7)^2+(-1)^2}\\ &=\sqrt{49+1}\\ &=\sqrt{50}\\ & = 5\sqrt{2} \end{align*}$$

\]

Step5: Calculate the perimeter

The perimeter \(P\) of \(\triangle ABD\) is \(AB + AD+BD\). Substituting the values:
\[

$$\begin{align*} P&=5 + 5+5\sqrt{2}\\ &=10 + 5\sqrt{2} \end{align*}$$

\]

Answer:

A. \(5\sqrt{2}+10\)