Let's analyze the given quadrilateral ABCD with vertices at [tex]\( A(2, 2) \)[/tex], [tex]\( B(2, 5) \)[/tex], [tex]\( C(7, 2) \)[/tex], and [tex]\( D(7, 5) \)[/tex] to determine its shape and area.
1. Determine the sides of the quadrilateral:
- Calculate the length of side [tex]\( AB \)[/tex] using the distance formula:
[tex]\[
AB = \sqrt{(2-2)^2 + (5-2)^2} = \sqrt{0 + 9} = 3
\][/tex]
- Calculate the length of side [tex]\( BC \)[/tex] (which is the same as side [tex]\( CD \)[/tex] and side [tex]\( DA \)[/tex]):
[tex]\[
BC = CD = DA = \sqrt{(7-2)^2 + (2-2)^2} = \sqrt{25 + 0} = 5
\][/tex]
2. Determine if opposite sides are equal:
- [tex]\( AB = CD = 3 \)[/tex]
- [tex]\( BC = DA = 5 \)[/tex]
Since opposite sides are equal and all angles are right angles (90 degrees), we can conclude that the quadrilateral is a rectangle.
3. Calculate the area of the rectangle:
- The area of a rectangle is given by length multiplied by width.
- Here, the length is [tex]\( AB = 3 \)[/tex] and the width is [tex]\( BC = 5 \)[/tex]:
[tex]\[
\text{Area} = \text{length} \times \text{width} = 3 \times 5 = 15 \text{ square units}
\][/tex]
Based on this analysis, we can conclude:
Quadrilateral [tex]\(ABCD\)[/tex] is a rectangle with an area of [tex]\(15\)[/tex] square units.
