Chapter 6.6, Problem 2CYP

To determine

To calculate: To show that QRST is a trapezoid and determine whether QRST is an isosceles trapezoid

Answer to Problem 2CYP

Quadrilateral QRST is a trapezium, but not an isosceles trapezoid

Explanation of Solution

Given: Quadrilateral QRST has vertices $Q\left(-8,-4\right),R\left(0,8\right),S\left(6,8\right),T\left(-6,-10\right)$

Formula Used:

Slope of line joining two points $\left(x_1,y_1\right)$ and $\left(x_2,y_2\right)$ is calculated as follows:

  $Slope=\frac{y_2-y_1}{x_2-x_1}$

When slope of two lines are equal, then the lines are parallel to each other

When product of slope of two lines is equal to $-1$ , then the lines are perpendicular to each other.

Quadrilateral is a trapezium if exactly one pair of opposite sides are parallel

Calculation:

Given a quadrilateral QRST has vertices $Q\left(-8,-4\right),R\left(0,8\right),S\left(6,8\right),T\left(-6,-10\right)$

Slope of sides of quadrilateral are calculated as follows:

  $m_{QR}=\frac{8-\left(-4\right)}{0-\left(-8\right)}=\frac{8+4}{8}=\frac{12}{8}=\frac{3}{2}$

  $m_{RS}=\frac{8-8}{6-0}=\frac{0}{6}=0=0$

  $m_{ST}=\frac{-10-8}{-6-6}=\frac{-18}{-12}=\frac{3}{2}$

  $m_{TQ}=\frac{-4-\left(-10\right)}{-8-\left(-6\right)}=\frac{-4+10}{-8+6}=\frac{6}{-2}=-3$

Since slope of sides $QR$ and $ST$ are equal

Thus, sides $QR$ and $ST$ are parallel to each other

In quadrilateral QRST, since exactly one pair of opposite sides are parallel, thus the quadrilateral is a trapezium

Now, using distance formula, let’s compare the length of non-parallel sides $RS$ and $TQ$

  $RS=\sqrt{{\left(0\right)}^2+{\left(6\right)}^2}=\sqrt{0+6^2}=\sqrt{36}$

  $TQ=\sqrt{{\left(6\right)}^2+{\left(-2\right)}^2}=\sqrt{36+4}=\sqrt{40}$

Since the length of sides are not equal, thus trapezium is not an isosceles trapezoid

Conclusion:

Hence, quadrilateral QRST is a trapezium, but not an isosceles trapezoid