Question

prove that quadrilateral pqrs is a trapezoid. determine whether quadrilateral pqrs is an isosceles trapezoid. show your work or explain how you determined your answer. enter your answer and your work or explanation in the space provided

Explanation:

Step1: Recall the slope - formula

The slope formula between two points $(x_1,y_1)$ and $(x_2,y_2)$ is $m=\frac{y_2 - y_1}{x_2 - x_1}$.

Step2: Calculate the slope of $PQ$

For points $P(4,10)$ and $Q(10,7)$, $m_{PQ}=\frac{7 - 10}{10 - 4}=\frac{- 3}{6}=-\frac{1}{2}$.

Step3: Calculate the slope of $QR$

For points $Q(10,7)$ and $R(6,4)$, $m_{QR}=\frac{4 - 7}{6 - 10}=\frac{-3}{-4}=\frac{3}{4}$.

Step4: Calculate the slope of $RS$

For points $R(6,4)$ and $S(4,5)$, $m_{RS}=\frac{5 - 4}{4 - 6}=\frac{1}{-2}=-\frac{1}{2}$.

Step5: Calculate the slope of $SP$

For points $S(4,5)$ and $P(4,10)$, the $x$ - coordinates are the same, so the slope of $SP$ is undefined (vertical line).

Step6: Recall the properties of a trapezoid

A trapezoid has one pair of parallel sides. Since $m_{PQ}=m_{RS}=-\frac{1}{2}$, $PQ\parallel RS$. So, quadrilateral $PQRS$ is a trapezoid.

Step7: Recall the distance - formula

The distance formula between two points $(x_1,y_1)$ and $(x_2,y_2)$ is $d=\sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}$.

Step8: Calculate the length of $PQ$

$d_{PQ}=\sqrt{(10 - 4)^2+(7 - 10)^2}=\sqrt{36 + 9}=\sqrt{45}=3\sqrt{5}$.

Step9: Calculate the length of $RS$

$d_{RS}=\sqrt{(4 - 6)^2+(5 - 4)^2}=\sqrt{4 + 1}=\sqrt{5}$.
Since $d_{PQ}
eq d_{RS}$, quadrilateral $PQRS$ is not an isosceles trapezoid.

Answer:

Quadrilateral $PQRS$ is a trapezoid but not an isosceles trapezoid.