Triangle JKL, with vertices J(-7,-8), K(-4,-4), and L(-8,-2), is drawn inside a rectangle, as shown below.What is the area, in square units, of triangle JKL? A=
units, squared units
2
The formula to find the area of a triangle is 1/2 * base * height.
To find the base, we can use the distance formula to find the length of JK.
JK = sqrt[(x2-x1)^2 + (y2-y1)^2]
JK = sqrt[(-4-(-7))^2 + (-4-(-8))^2]
= sqrt[(3)^2 + (4)^2]
= sqrt[9 + 16]
= sqrt[25]
= 5
To find the height, we can use the distance formula to find the length of the altitude from point L to JK.
The slope of JK is (change in y)/(change in x) = (-4+8)/(-4+7) = 4/3
The slope of the altitude is the negative reciprocal of the slope of JK: -3/4
Using the point-slope formula, we can find the equation of the altitude:
(y - y1) = m(x - x1)
(y - (-2)) = (-3/4)(x - (-8))
y + 2 = (-3/4)(x + 8)
y + 2 = (-3/4)(x) - 6
y = (-3/4)x - 8
To find the point where the altitude intersects JK, we can solve the system of equations:
y = (-3/4)x - 8
y = (4/3)x + b
The coordinates of the intersection point are the solution to this system, so we solve for x.
(-3/4)x - 8 = (4/3)x + b
(-3/4)x - (4/3)x = b + 8
(-9/12)x - (16/12)x = b + 8
(-25/12)x = b + 8
(-12/25)(-25/12)x = (-12/25)(b + 8)
x = (12/25)(8 - b)
We know that this point is on JK, so we can substitute for x and solve for y.
5 = sqrt[(12/25)(8 - b)]^2 + (-4)
5 = (12/25)(8 - b) + 16
5 = (96/25) - (12/25)b + 16
-11 = (12/25)b
-275 = 12b
-275/12 = b
So the coordinates of the intersection point are (x, y) = [ (12/25)(8 - (-275/12)), -275/12 ]
To find the height, we can use the distance formula to find the length of the altitude from point L to JK.
The distance formula is:
[LK] = sqrt[(x2-x1)^2 + (y2-y1)^2]
[LK] = sqrt[(-7-[(12/25)(8 - (-275/12))])^2 + (-8-(-275/12))^2]
[LK] = sqrt[(-7-[(96 - 275)/25])^2 + (-8-(-275/12))^2]
[LK] = sqrt[-7-[-419/25])^2 + (-96/12-(-275/12))^2]
[LK] = sqrt[-7 - (-419/25))^2 + (-96 - (-275/12))^2]
[LK] = sqrt[-7 + (419/25))^2 + (-96 + (275/12))^2]
[LK] = sqrt[(1925 + 419)/625)^2 + (-1152 + 275)/12))^2]
[LK] = sqrt[(2344/625)^2 + (-877/12)^2]
Now that we know the length of JK is 5 and the length of LK is sqrt[(2344/625)^2 + (-877/12)^2], we can find the area of triangle JKL.
Area = 1/2 * base * height
Area = 1/2 * 5 * sqrt[(2344/625)^2 + (-877/12)^2]
Area = 5/2 * sqrt[(2344/625)^2 + (-877/12)^2]
So the area of triangle JKL is 5/2 * sqrt[(2344/625)^2 + (-877/12)^2] square units.