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Two sides of a triangles are 5 and 5, and the inradius is 4/3.  Find the length of the third side.

 Aug 18, 2020
 #1
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Dragan: Please look at this question and see if you can solve it. I know the answer but don't know how to solve it. Thanks.  https://web2.0calc.com/questions/geometry_24827/new

 Aug 19, 2020
 #2
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a=5

b = ?

 

[(5*b - b^2/2)]  /  [2*(sqrt(25 - (b^2/4)))] =4/3, solve for b

 

Solve for b:
(5 b - b^2/2)/(2 sqrt(25 - b^2/4)) = 4/3

Cross multiply:
3 (5 b - b^2/2) = 8 sqrt(25 - b^2/4)

3 (5 b - b^2/2) = 8 sqrt(25 - b^2/4) is equivalent to 8 sqrt(25 - b^2/4) = 3 (5 b - b^2/2):
8 sqrt(25 - b^2/4) = 3 (5 b - b^2/2)

Raise both sides to the power of two:
64 (25 - b^2/4) = 9 (5 b - b^2/2)^2

Expand out terms of the left hand side:
1600 - 16 b^2 = 9 (5 b - b^2/2)^2

Expand out terms of the right hand side:
1600 - 16 b^2 = (9 b^4)/4 - 45 b^3 + 225 b^2

Subtract (9 b^4)/4 - 45 b^3 + 225 b^2 from both sides:
-(9 b^4)/4 + 45 b^3 - 241 b^2 + 1600 = 0

The left hand side factors into a product with five terms:
-1/4 (b - 10) (b - 8) (9 b^2 - 18 b - 80) = 0

Multiply both sides by -4:
(b - 10) (b - 8) (9 b^2 - 18 b - 80) = 0

Split into three equations:
b - 10 = 0 or b - 8 = 0 or 9 b^2 - 18 b - 80 = 0

Add 10 to both sides:
b = 10 or b - 8 = 0 or 9 b^2 - 18 b - 80 = 0

Add 8 to both sides:
b = 10 or b = 8 or 9 b^2 - 18 b - 80 = 0

Divide both sides by 9:
b = 10 or b = 8 or b^2 - 2 b - 80/9 = 0

Add 80/9 to both sides:
b = 10 or b = 8 or b^2 - 2 b = 80/9

Add 1 to both sides:
b = 10 or b = 8 or b^2 - 2 b + 1 = 89/9

Write the left hand side as a square:
b = 10 or b = 8 or (b - 1)^2 = 89/9

Take the square root of both sides:
b = 10 or b = 8 or b - 1 = sqrt(89)/3 or b - 1 = -sqrt(89)/3

Add 1 to both sides:
b = 10 or b = 8 or b = 1 + sqrt(89)/3 or b - 1 = -sqrt(89)/3

Add 1 to both sides:
b = 10 or b = 8 or b = 1 + sqrt(89)/3 or b = 1 - sqrt(89)/3

(5 b - b^2/2)/(2 sqrt(25 - b^2/4)) ⇒ (5 8 - 8^2/2)/(2 sqrt(25 - 8^2/4)) = 4/3
4/3 ⇒ 4/3 ≈ 1.33333:
So this solution is correct

(5 b - b^2/2)/(2 sqrt(25 - b^2/4)) ⇒ (5 10 - 10^2/2)/(2 sqrt(25 - 10^2/4)) = (undefined)
4/3 ⇒ 4/3 ≈ 1.33333:
So this solution is incorrect

(5 b - b^2/2)/(2 sqrt(25 - b^2/4)) ≈ -1.33333
4/3 ⇒ 4/3 ≈ 1.33333:
So this solution is incorrect

(5 b - b^2/2)/(2 sqrt(25 - b^2/4)) ≈ 1.33333
4/3 ⇒ 4/3 ≈ 1.33333:
So this solution is correct

The solution is:

 b = 8 - The 3rd side of isosceles triangle.

 Aug 19, 2020

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