Triangle DEF, with vertices D(-9,4), E(-2,2), and F(-3,7), is drawn inside a rectangle, as shown below.
What is the area, in square units, of triangle DEF?
To find the area of triangle DEF, we can use the formula for the area of a triangle:
A = (1/2) * base * height
To find the base, we can calculate the distance between points D and E, which gives us the length of the rectangle that contains triangle DEF. Using the distance formula:
base = sqrt((x2 - x1)^2 + (y2 - y1)^2)
base = sqrt((-2 - (-9))^2 + (2 - 4)^2)
= sqrt((7)^2 + (-2)^2)
= sqrt(49 + 4)
= sqrt(53)
The height of the triangle is the distance between point D and the line segment EF. To find this distance, we need to find the equation of the line containing line segment EF, and then substitute the x-coordinate of point D into the equation to find the y-coordinate.
The equation of the line containing line segment EF can be found using the slope-intercept form:
y - y1 = m(x - x1)
Given points E(-2, 2) and F(-3, 7), we can use the formula for slope:
m = (y2 - y1) / (x2 - x1)
= (7 - 2) / (-3 - (-2))
= 5 / (-1)
= -5
Using point E(-2, 2) and the slope, we can write the equation of the line in slope-intercept form:
y - 2 = -5(x - (-2))
y - 2 = -5(x + 2)
y - 2 = -5x - 10
y = -5x - 8
Now we substitute the x-coordinate of point D into the equation to find the y-coordinate:
y = -5(-9) - 8
y = 45 - 8
y = 37
Therefore, the height of the triangle is 37 units.
Now we can find the area of the triangle:
A = (1/2) * base * height
= (1/2) * sqrt(53) * 37
= (1/2) * 37 * sqrt(53)
= 37/2 * sqrt(53)
So the area of triangle DEF is 37/2 * sqrt(53) square units.