Question

find the area of △cde. round the area to the nearest whole number, if necessary.
a(-5, 4)
4
b(0, 3)
2
f(-2, 1)
-4 -2 2 4 6 x
-2
c(4, -1)
e(2, -3)
-6
d(4, -5)
the area is square units

Explanation:

Step1: Identify coordinates of C, D, E

C(4, -1), D(4, -5), E(2, -3)

Step2: Find length of CD (vertical side)

Since x-coordinates of C and D are same, \( CD = |y_D - y_C| = |-5 - (-1)| = |-4| = 4 \)

Step3: Find horizontal distance from E to CD (base)

CD is vertical line x=4. Horizontal distance from E(2, -3) to x=4 is \( |4 - 2| = 2 \)

Step4: Calculate area of triangle

Area of triangle = \( \frac{1}{2} \times base \times height = \frac{1}{2} \times 4 \times 2 = 4 \) Wait, no, wait. Wait, maybe better to use shoelace formula. Let's recalculate with shoelace.

Coordinates: C(4, -1), D(4, -5), E(2, -3)

Shoelace formula: \( \frac{1}{2} |x_1(y_2 - y_3) + x_2(y_3 - y_1) + x_3(y_1 - y_2)| \)

Plug in: \( x_1=4, y_1=-1; x_2=4, y_2=-5; x_3=2, y_3=-3 \)

\( \frac{1}{2} |4(-5 - (-3)) + 4(-3 - (-1)) + 2(-1 - (-5))| \)

\( = \frac{1}{2} |4(-2) + 4(-2) + 2(4)| \)

\( = \frac{1}{2} |-8 -8 +8| = \frac{1}{2} |-8| = 4 \)? Wait, that can't be right. Wait, maybe I messed up the coordinates. Wait, CD is from (4,-1) to (4,-5), so length is 4. The base is CD, length 4. The height is the horizontal distance from E to CD, which is 4 - 2 = 2. So area is 0.542=4? Wait, but let's check the graph. Alternatively, maybe the base is CE or DE. Wait, no, let's plot the points. C(4,-1), D(4,-5) is vertical line down 4 units. E(2,-3) is to the left of CD. So the triangle has vertices at (4,-1), (4,-5), (2,-3). So the base is CD (length 4), and the height is the horizontal distance from E to x=4, which is 2. So area is 0.542=4. Wait, but maybe I made a mistake. Wait, another way: use coordinates to find vectors or base and height. Wait, CD is vertical, length 4. The horizontal distance from E to CD is 2 (since E is at x=2, CD at x=4). So area is 0.5baseheight = 0.542=4. Wait, but let's check with shoelace again.

Shoelace formula:

List the points in order: C(4, -1), D(4, -5), E(2, -3), back to C(4, -1)

Sum1 = (4(-5)) + (4(-3)) + (2*(-1)) = -20 -12 -2 = -34

Sum2 = (-14) + (-52) + (-3*4) = -4 -10 -12 = -26

Area = 0.5|Sum1 - Sum2| = 0.5|-34 - (-26)| = 0.5|-8| = 4. So yes, area is 4. Wait, but that seems small. Wait, maybe I misread the coordinates. Let's check the graph again. C is (4, -1), D is (4, -5), so vertical distance is 4. E is (2, -3). So the triangle is between x=2, x=4, y=-5 to y=-1. So the base is 4 (vertical), height is 2 (horizontal). So area is 4. Alternatively, maybe the base is DE or CE. Wait, DE: distance between D(4,-5) and E(2,-3): \( \sqrt{(4-2)^2 + (-5+3)^2} = \sqrt{4 +4} = \sqrt{8} \). CE: distance between C(4,-1) and E(2,-3): \( \sqrt{(4-2)^2 + (-1+3)^2} = \sqrt{4 +4} = \sqrt{8} \). CD: 4. So it's an isoceles triangle with two sides \( \sqrt{8} \) and base 4. Then height h: \( h^2 + (4/2)^2 = (\sqrt{8})^2 \) => \( h^2 +4 =8 \) => \( h^2=4 \) => h=2. Then area is 0.542=4. Yep, that matches. So the area is 4. Wait, but maybe I made a mistake. Wait, the graph: C is at (4, -1), D at (4, -5), E at (2, -3). So the triangle is formed by these three points. So the base CD is 4 units (vertical), and the height is the horizontal distance from E to CD, which is 2 units (since E is at x=2, CD at x=4). So area is 0.54*2=4. So the area is 4.

Answer:

4