Question

1. determine if the given points are vertices of a trapezoid. a(1, 9), b(4, 2), c(5, 2), d(8, 9). a. prove that abcd is a trapezoid. show all steps! b. is abcd an isosceles trapezoid? why or why not? show all work!

Explanation:

Step1: Recall Trapezoid Definition

A trapezoid is a quadrilateral with at least one pair of parallel sides. Parallel sides have equal slopes. The slope formula is \( m = \frac{y_2 - y_1}{x_2 - x_1} \).

Step2: Calculate Slope of AB

For points \( A(1, 9) \) and \( B(4, 2) \):
\( m_{AB} = \frac{2 - 9}{4 - 1} = \frac{-7}{3} = -\frac{7}{3} \).

Step3: Calculate Slope of BC

For points \( B(4, 2) \) and \( C(5, 2) \):
\( m_{BC} = \frac{2 - 2}{5 - 4} = \frac{0}{1} = 0 \).

Step4: Calculate Slope of CD

For points \( C(5, 2) \) and \( D(8, 9) \):
\( m_{CD} = \frac{9 - 2}{8 - 5} = \frac{7}{3} \).

Step5: Calculate Slope of DA

For points \( D(8, 9) \) and \( A(1, 9) \):
\( m_{DA} = \frac{9 - 9}{1 - 8} = \frac{0}{-7} = 0 \).

Step6: Identify Parallel Sides

Compare slopes:

• \( m_{BC} = 0 \) and \( m_{DA} = 0 \), so \( BC \parallel DA \).

Since there is at least one pair of parallel sides, \( ABCD \) is a trapezoid.

Part (b) Explanation:

Step1: Recall Isosceles Trapezoid Definition

An isosceles trapezoid has congruent non - parallel sides (legs) and congruent base angles. Also, the distance between the parallel sides (bases) and the length of the legs can be used. Alternatively, in a trapezoid, if the non - parallel sides (legs) are congruent, it is isosceles. First, find the lengths of the non - parallel sides (AB and CD).

Step2: Calculate Length of AB

Using the distance formula \( d=\sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2} \) for \( A(1,9) \) and \( B(4,2) \):
\( AB=\sqrt{(4 - 1)^2+(2 - 9)^2}=\sqrt{3^2+(-7)^2}=\sqrt{9 + 49}=\sqrt{58} \).

Step3: Calculate Length of CD

Using the distance formula for \( C(5,2) \) and \( D(8,9) \):
\( CD=\sqrt{(8 - 5)^2+(9 - 2)^2}=\sqrt{3^2+7^2}=\sqrt{9 + 49}=\sqrt{58} \).

Step4: Check for Isosceles Trapezoid

Since the lengths of the non - parallel sides \( AB \) and \( CD \) are equal (\( AB = CD=\sqrt{58} \)), \( ABCD \) is an isosceles trapezoid.

Answer:

a. \( ABCD \) is a trapezoid because \( BC \parallel DA \) (slopes of \( BC \) and \( DA \) are both 0).
b. \( ABCD \) is an isosceles trapezoid because the non - parallel sides \( AB \) and \( CD \) have equal length (\( \sqrt{58} \)).