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

Question

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

Sadie Adams · Accepted Answer

#" "#
Radius of the Triangle's Inscribed Circle #= 1.28# Units

#" "#
A triangle has vertices at #(4,1), (2,6) and (7,3)#.

Plot the points on a Cartesian Coordinate Plane and label them as #A, B and C# respectively.

Measure the magnitudes of the line segments #bar (AB), bar (BC) and bar (AC)#

Perimeter of the triangle #AB + BC + AC#

#rArr 5.39 + 5.83 + 3.61#

#rArr 14.83# Units

Semi-Perimeter [ s ] #=(Perimeter)/2#

#rArr 14.83/2# Units

#rArr 7.415# Units

Next, construct angle bisectors..

These three angle bisectors intersect at a point called Incenter.

Using the Incenter as one point and the three sides #AB, BC and AC#, construct perpendicular lines and mark the points where they intersect the sides of the triangle.

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 #1.28# units.

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

#r^2 = [ (s-a)(s-b)(s-c) ]/s#

#rArr r^2=[(7.415-5.83)(7.415-3.61)(7.415-5.39)]/7.415#

#rArr r^2 = 12.21262313/7.415#

#rArr r^2 = 1.647015931#

#rArr r = sqrt(1.647015931)#

#rArr r ~~ 1.283361185#

#r~~1.28 # Units.

Hence, Radius of the Inscribed circle #~~1.28# units.

Hope it helps.

Wyatt Anderson · Answer

undefined 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.