# A triangle has corners at #(5 , 2 )#, #(2 ,3 )#, and #(3 ,4 )#. What is the radius of the triangle's inscribed circle?

From diagram:

They also all have heights

So it follows:

Let:

Using the distance formula:

Now we need to find area of

There are different ways we can do this. Having the lengths of the sides we could use Heron's formula, or we could find the height of ABC using line equations. Since we have surds for the lengths of the sides, Heron's formula will be pretty messy and a calculator or computer will be really helpful.

Heron's formula is given as:

Where

I won't include the calculation for this, I will just give the result.

So to find radius:

PLOT:

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To find the radius of the triangle's inscribed circle, we can use the formula:

radius = (Area of the triangle) / (Semiperimeter of the triangle)

First, we need to calculate the area of the triangle using the coordinates given. We can use the formula for the area of a triangle given its vertices:

Area = 1/2 * |x1(y2 - y3) + x2(y3 - y1) + x3(y1 - y2)|

Then, we calculate the semiperimeter of the triangle, which is the sum of the lengths of its three sides divided by 2:

Semiperimeter = (side1 + side2 + side3) / 2

Finally, we can use the formula for the radius of the inscribed circle:

radius = Area / Semiperimeter

By substituting the coordinates of the vertices into the formulas, we can calculate the radius of the inscribed circle.

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When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

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- Points #(2 ,1 )# and #(5 ,9 )# are #(3 pi)/4 # radians apart on a circle. What is the shortest arc length between the points?
- A triangle has corners at #(2 ,8 )#, #(3 ,9 )#, and #(4 ,7 )#. What is the area of the triangle's circumscribed circle?
- A triangle has vertices A, B, and C. Vertex A has an angle of #pi/12 #, vertex B has an angle of #(5pi)/8 #, and the triangle's area is #8 #. What is the area of the triangle's incircle?

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