# Use calculus to find the area of the triangle with the vertices (0, 5), (2, -2), and (5, 1).?

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B(2,-2)

C(5,1)

draw the triangle ABC on a graph paper.

Plot the point D(5,5)

From A, draw a horizontal line AD (parallel to x-axis)

Plot the point F(0, -2)

plot the point E(5, -2)

from B, draw a horizontal line (FBE)

calculate the

area of the reactangle (AFED) =7*5=35

area of triangle (AFB) = .5 * 7*2 = 7

area of triangle (BCE) = .5 * 3*3 = 4.5

area of triangle (ADC) = .5 * 5*4 = 10

so,

area triangle (ABC) = area (AFED) – area(AFB) – area (BCE) – area(ADC)

area triangle (ABC) = 35 – 7 – 4.5 – 10 = 13.5

Here’re some steps:

1. You’re given 3 vertices, call them A, B, and C.

2. Find the equations of the 3 lines AB, BC, and CA.

How do you do that?

If A is (x1, y1) and B is (x2, y2), equation of line going thru A and B is y – y1 = m * ( x – x1) , where m = (y2 – y1)/(x2 – x1)

3. Once you get the equations of AB, BC, and CA, plot the lines on a graph, you should get a triangle

4. At this point, you should figure out if your variable of integration is dx or dy and integrate accordingly to get the area covered by the lines. I am assuming you know how to carry forward from here.

Hope this helps 🙂

Mahurshi Akilla

The points then become: A(0,7), B(2,0), C(5,3).

If you work out the equations of the lines relative to the new axes, you can then integrate these equations to find the areas:

P bounded by AB, the x axis, and the y axis;

Q bounded by BC, the x axis, and the line x = 5;

R bounded by AC, the x axis, the y axis, and the line x = 5.

The area of the triangle is then R – (P + Q).

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