The area of a triangle ABC is 126 sq.cm.

AC=13 cm , BC = 21 cm, what is the length of AB?
where, triangle ABC is a scalene triangle

I'd use Heron's formula. Here, if c is the length of AB,

s = (34+c)/2 = 17 + c/2

so we have

(17+c/2)(4+c/2)(-4+c/2)(17-c/2) = 126^2

It looks messy, but expands into a nice simple quadratic in c^2.

c=20

Introduce values triancal.esy.es/?a=21&b=13&t=126

c = 20 or
c = 28.6356421265
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To find the length of side AB in a scalene triangle, we can use the formula for the area of a triangle:

Area = (1/2) * base * height

In this case, the base of the triangle can be any one of the three sides. Let's choose AB as the base. The height of the triangle will be the perpendicular distance from the opposite vertex (C) to the base (AB).

Given that the area of the triangle is 126 sq.cm, we can substitute the values into the formula:

126 = (1/2) * AB * height

Now, we need to find the height of the triangle. To find the height, we can use the formula of a right-angled triangle:

a^2 + b^2 = c^2

In our case, we have AC = 13 cm and BC = 21 cm. The length of AB is what we want to find.

To find the height, we need to find the length of the segment from C perpendicular to AB. Let's call this length h.

Using the Pythagorean theorem, we have:

h^2 + AB^2 = AC^2
h^2 + AB^2 = 13^2

Similarly:

h^2 + BC^2 = AB^2
h^2 + 21^2 = AB^2

We now have two equations with two unknowns (h and AB). We can solve these equations simultaneously using substitution or elimination. Let's use substitution method.

From the first equation:
h^2 + AB^2 = 13^2
h^2 + AB^2 = 169

Simplifying and isolating h^2:
h^2 = 169 - AB^2

Now, we substitute this expression for h^2 in the second equation:

(169 - AB^2) + 21^2 = AB^2
169 - AB^2 + 441 = AB^2
610 = 2AB^2
305 = AB^2

Taking the square root of both sides:
√305 = AB

Therefore, the length of side AB is approximately 17.49 cm.