MHB Length of a third side of triangle

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To find the length of the third side of a triangle with an area of 18 and sides of lengths 5 and 10, multiple methods were discussed. The first method used the sine area formula and the law of cosines, yielding approximately 7.45. The second method applied Heron's formula, resulting in two potential lengths for the third side: approximately 14.775 and 5.63. The calculations involved in both methods were verified to ensure consistency, with the acute and obtuse angles providing valid solutions. Ultimately, the problem highlighted the complexity of triangle side lengths and the importance of using different mathematical approaches for verification.
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Math Problem: Find the length of the third side of a triangle if the area of the triangle is 18 and two of its sides have lengths of 5 and 10.

Which one of these are correct when I am working them out? If none of these are correct, then can somebody please help me solve this math problem step-by-step?

First way I worked out the problem:

A=18=0.5*5*10*sin(x)

x = 46 degrees

====

c^2 = 10^2+5^2-2*50*cos(46) = 55,53

third side:

c = 7,45

Second way I worked out the problem:

I know A = 18 units², a = 5 units, and b = 10 units

given A = (absin(C))/2

=> 18 = 25sin(C)

=> 18/25 = sin(C)

=> C = sin⁻¹(18/25)

given c² = a² + b² -2abcos(C)=> c² = 25 + 100 -100cos(sin⁻¹(18/25))

=> c = √(125 -100cos(sin⁻¹(18/25)))

=> c = √(125 - 4√301) units

=> c ≈ 7.4567 ( 4 dp)

Third way I worked out the problem:

Use Heron's formula for triangle:

Suppose the third side is x, and the others are 5 and 10, so by Heron formula, we get:

Area = sqrt(s(s-a)(s-b)(s-c))

where s=semi perimeter, a,b,c are sides of triangle, a=x, b=5, c=10

so, s=1/2.(x+5+10)

=1/2(x+15)

s-a= 1/2x +15/2 -x= 15/2-x/2

s-b=x/2+15/2-10/2=x/2+5/2

s-c=x/2+15/2-20/2=x/2-5/2

Area= sqrt(x/2+15/2)(15/2-x/2)(x/2+5/2)(x/2-5/…

324=(15/2+x/2)(15/2-x/2)(x/2+5/2)(x/2-…

324=(225/4 -x^2/4)(x^2/4-25/4)..multiply by 4 to get

1296=(225-x^2)(x^2-25)

225x^2-225*25-x^4+25x^2-1296=0

-x^4+250x^2-6921=0

-(x^4-250x^2+6921)=0

-((x^2-125)-8704)=0

(x^2-16sqrt34-125)(x^2+16sqrt34-125)=0

x^2=16sqrt34+125

x=sqrt(16sqrt 34+125)

=14.775

or

x^2=125-16sqrt34

x=sqrt(125-16sqrt34)

=5.63

So, the length of the third side is 14.775 or 5.63

I am really lost and confused on this problem.
 
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I would look at the two case:

View attachment 1340

In both cases, we have:

$$A=\frac{1}{2}bh$$

$$18=25\sin(\theta)$$

Case 1:

$$\theta=\pi-\sin^{-1}\left(\frac{18}{25} \right)$$

Using the law of cosines, we may write:

$$x=\sqrt{10^2+5^2-2\cdot5\cdot10\cos\left(\pi-\sin^{-1}\left(\frac{18}{25} \right) \right)}$$

Case 2:

$$\theta=\sin^{-1}\left(\frac{18}{25} \right)$$

Using the law of cosines, we may write:

$$x=\sqrt{10^2+5^2-2\cdot5\cdot10\cos\left(\sin^{-1}\left(\frac{18}{25} \right) \right)}$$

You should be able to obtain an exact value for $x$ in both cases (you have already found the exact value for the acute case), and these do agree with the two positive roots that Heron's formula gives.

What do you find?
 

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Hi,
It may be easier to assign coordinates in the problem. See the attachment:

View attachment 1342
 

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