Question

Derive the expression for centroid of right angled triangle

Answer 1

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Derive the expression for the centroid of right angled triangle.

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Along the hypotenuse, y = ax.

The area of the triangle is

The x coordinate of the centroid,X, multiplied by the area is equal to the integral of x multiplied by the area da. That is, it is the sum of the small areas da multiplied by their x coordinate.

where,

Therefore,

Solving for the x coordinate of the centroid, 

If you can show where the medians (the segment from a vertex to the midpoint of the other side of the right triangle) intersect each other, then this point of intersection is the CENTROID. To calculate this location, take 2/3 the distance from each vertex to the midpoint of the other side.

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Answer 2

Answer:

expression for centroid of right angled triangle

Step-by-step explanation:

Formula for the hypotenuse:

y = y(x) = a - (a/b)x

x = x(y) = b - (b/a)y

A = ba/2

xbar = (1/A)∫xdA

dA = ydx = [a - (a/b)x]dx

xbar = [2/(ba)] [ a∫xdx - (a/b)∫x²dx ] : where the integration is over 0 ≤ x < b

xbar = [2/(ba)] [ ab²/2 - ab²/3 ] = 2ab²/(6ab) = b/3

ybar = (1/A)∫ydA

dA = xdy = [b - (b/a)y]dy

ybar = [2/(ab)] [ b∫ydy - (b/a)∫y²dy ] : where the integration is over 0 ≤ y ≤ a

ybar = (2/(ab)] [ a²b/2 - (a²b/3) ] = 2a²b/(6ab) = a/3

Coordinates of the centroid: (xbar, ybar) = (b/3, a/3) where a and b are the altitude and base of the triangle respectively.