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What is the radius of the circle inscribed in triangle ABC if AB = 5, AC=7, BC=8? Express your answer in simplest radical form.

 Dec 24, 2020
 #1
avatar+859 
+3

What is the radius of the circle inscribed in triangle ABC if AB = 5, AC=7, BC=8? Express your answer in the simplest radical form.

 

r = √3        area of ΔABC = 10r         area of a circle is  3pi   wink

 Dec 24, 2020
 #2
avatar+114320 
+1

Let B  =(0,0)   C = (8,0)

And  we can find the x coordinate of  A by letting the  circles   x^2 + y^2  = 25   and  (x - 8)^2 + y^2 = 49  intersect

 

Subbing    y^2  = 25 -x^2  into  the second equation we have that

 

(x - 8)^2  + 25 - x^2  =  49

x^2  - 16x + 64  + 25 - x^2  = 49

-16x + 89  = 49

16x  = 40

x = 40/16  = 2.5  ⇒    x^2  = 6.25

And  y = sqrt (25 - 6.25)  = sqrt [ 18.75 ]   = sqrt (75/4)  = sqrt (25 * 3) / 2  = (5/2) sqrt (3)=  the altitude of this triangle

 

The area of this triangle  =   (1/2)( BC) (altitude) =  (1/2) (8) (5/2) sqrt (3)  = 10sqrt (3)

 

We can find the radius,R, of the  inscribed circle  thusly

 

(1/2) R  ( 8 + 7 + 5 )  =  10sqrt (3)

 

10R  =  10sqrt (3)

 

R =  sqrt (3)

 

cool cool cool

 Dec 25, 2020

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