Questions: a. What quadrilateral is ABCD? A(2,0) · C(7,7) B(10,0)=D(0,7) b. Prove it algebraically. c. Find the perimeter of the quadrilateral. d. Find the area of the triangle.

Solution Steps

The coordinates of the quadrilateral ABCD are given as:

• A(2, 0)
• B(10, 0)
• C(7, 7)
• D(0, 7)

To determine the type of quadrilateral, we need to check the lengths of the sides and the slopes of the sides to see if any sides are parallel or equal in length.

Using the distance formula \( d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \):

• \( AB = \sqrt{(10 - 2)^2 + (0 - 0)^2} = \sqrt{8^2} = 8 \)
• \( BC = \sqrt{(7 - 10)^2 + (7 - 0)^2} = \sqrt{(-3)^2 + 7^2} = \sqrt{9 + 49} = \sqrt{58} \)
• \( CD = \sqrt{(7 - 0)^2 + (7 - 7)^2} = \sqrt{7^2} = 7 \)
• \( DA = \sqrt{(0 - 2)^2 + (7 - 0)^2} = \sqrt{(-2)^2 + 7^2} = \sqrt{4 + 49} = \sqrt{53} \)

Using the slope formula \( m = \frac{y_2 - y_1}{x_2 - x_1} \):

• \( \text{slope of } AB = \frac{0 - 0}{10 - 2} = 0 \)
• \( \text{slope of } BC = \frac{7 - 0}{7 - 10} = -\frac{7}{3} \)
• \( \text{slope of } CD = \frac{7 - 7}{7 - 0} = 0 \)
• \( \text{slope of } DA = \frac{7 - 0}{0 - 2} = -\frac{7}{2} \)

From the slopes and lengths, we can see:

• \( AB \parallel CD \) (both have slope 0)
• \( AB \neq CD \) (lengths are different)
• \( BC \neq DA \) (lengths are different)
• No sides are equal in length

Since opposite sides are parallel but not equal in length, ABCD is a trapezoid.

Final Answer