# A triangle has corners at #(4 ,1 )#, #(2 ,6 )#, and #(7 ,3 )#. What is the radius of the triangle's inscribed circle?

Radius of the Triangle's Inscribed Circle

A triangle has vertices at

Plot the points on a Cartesian Coordinate Plane and label them as

Measure the magnitudes of the line segments

Perimeter of the triangle

Semi-Perimeter [ s ]

Next, construct angle bisectors..

These three angle bisectors intersect at a point called Incenter.

Using the Incenter as one point and the three sides

Measure the length of these lines from the Incenter

Construct a Circle, the center being the Incenter and one of the points on the sides as the Radius.

Note that all of them have the same magnitude

We can also use the formula given below to find the magnitude of the radius.

Hence, Radius of the Inscribed circle

Hope it helps.

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To find the radius of the inscribed circle of a triangle, you can use the formula:

[ r = \frac{2A}{P} ]

Where ( A ) is the area of the triangle and ( P ) is the perimeter of the triangle.

[ A = \frac{1}{2} \left| x_1(y_2 - y_3) + x_2(y_3 - y_1) + x_3(y_1 - y_2) \right| ]

[ P = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} + \sqrt{(x_3 - x_2)^2 + (y_3 - y_2)^2} + \sqrt{(x_3 - x_1)^2 + (y_3 - y_1)^2} ]

Substitute the coordinates of the given points into these formulas to calculate the area and perimeter of the triangle. Then, use these values to find 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.

- A circle's center is at #(2 ,7 )# and it passes through #(3 ,1 )#. What is the length of an arc covering #(2pi ) /3 # radians on the circle?
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- A triangle has corners at #(9 ,8 )#, #(2 ,3 )#, and #(7 ,4 )#. 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 #(3pi)/8 #, and the triangle's area is #21 #. What is the area of the triangle's incircle?
- A circle's center is at #(9 ,3 )# and it passes through #(2 ,6 )#. What is the length of an arc covering #(2pi ) /3 # radians on the circle?

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