Question

Payton cut out two shapes, as shown, that she will later put together to resemble a house. On a coordinate plane, a square and a triangle are shown. The square has points (1, 4), (4, 7), (7, 4), (4, 1). The triangle has points (5, 8), (7, 8), and (8, 5). What is the total area of the two shapes? 20 square units 21 square units 22 square units 23 square units

Answer 1

Answer:

Correct option: second one -> 21 square units

Step-by-step explanation:

First let's find the side of the square.

We can find this side finding the distance between two points of the square, using the formula:

$distance = \sqrt{(x_1 - x_2)^2 + (y_1 - y_2)^2}$

Using points (1,4) and (4,7), we have:

$distance = \sqrt{(1 - 4)^2 + (4 - 7)^2}$

$distance = \sqrt{(-3)^2 + (-3)^2}$

$distance = 3\sqrt{2}$

So the area of the square is:

$area = distance^2$

$area = (3\sqrt{2} )^2 = 18$

Now, let's find the three sides of the triangle:

points (5,8) and (7,8):

$distance = \sqrt{(5 - 7)^2 + (8 - 8)^2} = 2$

points (7,8) and (8,5):

$distance = \sqrt{(7 - 8)^2 + (8 - 5)^2} = \sqrt{10}=3.1623$

points (8,5) and (5,8):

$distance = \sqrt{(8 - 5)^2 + (5 - 8)^2} = 3\sqrt{2}=4.2426$

The area of the triangle can be calculated using the formula:

$area = \sqrt{p(p-a)(p-b)(p-c)}$

Where p is the semi perimeter of the triangle:

$p = (a+b+c)/2 = (2+\sqrt{10} +3\sqrt{2} )/2 = 4.7025$

So the area is:

$area = \sqrt{4.7025(4.7025-2)(4.7025-3.1623)(4.7025-4.2426)} = 3$

Now, adding both areas, we have:

Total area = 18 + 3 = 21 square units

Correct option: second one