Question

the points ( i(4,4) ), ( j(1,8) ), ( k(-3,5) ), and ( l(0,1) ) form a quadrilateral. find the desired slopes and lengths, then fill in the words that best identifies the type of quadrilateral.
answer attempt 1 out of 2
slope of ( overline{ij} = )
length of ( overline{ij} = )
slope of ( overline{jk} = )
length of ( overline{jk} = )
slope of ( overline{kl} = )
length of ( overline{kl} = )
slope of ( overline{li} = )
length of ( overline{li} = )
quadrilateral ( ijkl ) is

Explanation:

Step1: Calculate slope of $\overline{IJ}$

The slope formula is $m = \frac{y_2 - y_1}{x_2 - x_1}$. For points $I(4,4)$ and $J(1,8)$, we have $x_1 = 4, y_1 = 4, x_2 = 1, y_2 = 8$. So slope $m_{IJ} = \frac{8 - 4}{1 - 4} = \frac{4}{-3} = -\frac{4}{3}$.

Step2: Calculate length of $\overline{IJ}$

The distance formula is $d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$. For $I(4,4)$ and $J(1,8)$, $d_{IJ} = \sqrt{(1 - 4)^2 + (8 - 4)^2} = \sqrt{(-3)^2 + 4^2} = \sqrt{9 + 16} = \sqrt{25} = 5$.

Step3: Calculate slope of $\overline{JK}$

For points $J(1,8)$ and $K(-3,5)$, $x_1 = 1, y_1 = 8, x_2 = -3, y_2 = 5$. Slope $m_{JK} = \frac{5 - 8}{-3 - 1} = \frac{-3}{-4} = \frac{3}{4}$.

Step4: Calculate length of $\overline{JK}$

Using distance formula for $J(1,8)$ and $K(-3,5)$, $d_{JK} = \sqrt{(-3 - 1)^2 + (5 - 8)^2} = \sqrt{(-4)^2 + (-3)^2} = \sqrt{16 + 9} = \sqrt{25} = 5$.

Step5: Calculate slope of $\overline{KL}$

For points $K(-3,5)$ and $L(0,1)$, $x_1 = -3, y_1 = 5, x_2 = 0, y_2 = 1$. Slope $m_{KL} = \frac{1 - 5}{0 - (-3)} = \frac{-4}{3} = -\frac{4}{3}$.

Step6: Calculate length of $\overline{KL}$

Using distance formula for $K(-3,5)$ and $L(0,1)$, $d_{KL} = \sqrt{(0 - (-3))^2 + (1 - 5)^2} = \sqrt{3^2 + (-4)^2} = \sqrt{9 + 16} = \sqrt{25} = 5$.

Step7: Calculate slope of $\overline{LI}$

For points $L(0,1)$ and $I(4,4)$, $x_1 = 0, y_1 = 1, x_2 = 4, y_2 = 4$. Slope $m_{LI} = \frac{4 - 1}{4 - 0} = \frac{3}{4}$.

Step8: Calculate length of $\overline{LI}$

Using distance formula for $L(0,1)$ and $I(4,4)$, $d_{LI} = \sqrt{(4 - 0)^2 + (4 - 1)^2} = \sqrt{4^2 + 3^2} = \sqrt{16 + 9} = \sqrt{25} = 5$.

Step9: Identify the quadrilateral

Since both pairs of opposite sides have equal slopes (so they are parallel) and equal lengths, the quadrilateral is a rhombus (also a parallelogram, but since all sides are equal, it's a rhombus).

Answer:

slope of $\overline{IJ}$ = $-\frac{4}{3}$
length of $\overline{IJ}$ = $5$
slope of $\overline{JK}$ = $\frac{3}{4}$
length of $\overline{JK}$ = $5$
slope of $\overline{KL}$ = $-\frac{4}{3}$
length of $\overline{KL}$ = $5$
slope of $\overline{LI}$ = $\frac{3}{4}$
length of $\overline{LI}$ = $5$
Quadrilateral $IJKL$ is a rhombus (or parallelogram, but rhombus is more precise as all sides are equal)