Question

PQRSTU is plotted on a coordinate plane with vertices p(-1,0) Q(-1,3) R(1,4) S(2,5) T(4,4) and U(4,0) what is the hexagons area

Answer 1

To find the area of the hexagon, we can split it into two triangles and a rectangle, and then add up the areas of each shape.

First, we can find the length and width of the rectangle by finding the distance between points R and T, and between points Q and U, respectively.

Length of the rectangle RT = sqrt((4-1)^2 + (4-5)^2) = sqrt(10)

Width of the rectangle QU = sqrt((4-(-1))^2 + (0-3)^2) = sqrt(50)

The area of the rectangle is then:

Area of rectangle = Length x Width = sqrt(10) x sqrt(50) = 10sqrt(2)

To find the area of the two triangles, we can use the formula for the area of a triangle:

Area of triangle = 1/2 x Base x Height

Triangle 1 has base PQ, which has length 3, and height 1, which is the distance between PQ and RS. The area of triangle 1 is:

Area of triangle 1 = 1/2 x 3 x 1 = 3/2

Triangle 2 has base RS, which has length sqrt((4-1)^2 + (5-4)^2) = sqrt(10), and height 1, which is the distance between PQ and RS. The area of triangle 2 is:

Area of triangle 2 = 1/2 x sqrt(10) x 1 = sqrt(10)/2

Therefore, the total area of the hexagon is:

Area of hexagon = Area of rectangle + Area of triangle 1 + Area of triangle 2
= 10sqrt(2) + 3/2 + sqrt(10)/2
= 10sqrt(2) + (3+sqrt(10))/2

So the area of the hexagon is approximately 19.55 square units.