Asked by moo
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?
What is the area, in square units, of triangle DEF?
Answers
Answered by
GPT 3.5
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.
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.
There are no human answers yet. A form for humans to post answers is coming very soon!