The other side lengths of the triangle are 26 and 28 cm
How to determine the other lengths?
The perimeter (P) is given as:
P = 84 cm
The area is given as:
A = 336
Let the sides be x, y and z.
So, we have:
x + y + z = 84
By herons' formula, we have:
[tex]Area = \sqrt{s(s-x)(s-y)(s-z)}[/tex]
Where:
s = 0.5(x + y + z)
Multiply by 2
2s = x + y + z
Recall that:
x + y + z = 84
So, we have:
2s = 84
Divide by 2
s = 42
Let x = 30 ---- the given side length
So, we have:
30 + y + z = 84
Subtract 30 from both sides
y + z = 54
Make y the subject
y = 54 - z
Recall that
[tex]Area = \sqrt{s(s-x)(s-y)(s-z)}[/tex]
The area is 336. So, we have:
[tex]336 = \sqrt{s(s-x)(s-y)(s-z)}[/tex]
Square both sides
112896 = s(s-x)(s-y)(s-z)
Substitute values for x, y and s
112896 = 42(42-30)(42 - (54 - z))(42 - z)
Divide through by 42
2688 = (42-30)(42 - (54 - z))(42 - z)
Divide through by 12
224 = (42 - (54 - z))(42 - z)
Evaluate the brackets
224 = (z - 12)(42 - z)
Using a graphing calculator, we have:
z =26 or z = 28
Recall that:
y = 54 - z
So, we have
y = 54 - 26 = 28
y = 54 - 28 = 26
Hence, the other side lengths of the triangle are 26 and 28 cm
Read more about triangles at:
https://brainly.com/question/1675117
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