Question

GEOMETRY Heron's formula states that the area of a triangle whose sides have lengths

1
a, b, and c is A = √s(s-a)(s - b)(s-c) where s =1/2 (a + b + c). If the area of
the triangle is 270 cm², s = 45 cm, a = 15 cm, and c = 39 cm, what is the length of side
b?

Answer 1

Approximately 26.87 cm

Explanation:

To find the length of side b using Heron's formula, we need to substitute the given values into the formula and solve for b:

A = √s(s-a)(s - b)(s-c)

270 = √45(45-15)(45-b)(45-39)

Simplifying the expression inside the square root:

270 = √45(30)(6)(6-b)

270 = 540√(6-b)

Squaring both sides:

72900 = 291600 - 54000b + 3240b^2

Rearranging:

3240b^2 - 54000b + 218700 = 0

Dividing both sides by 540:

6b^2 - 100b + 405 = 0

We can solve for b using the quadratic formula:

b = (-(-100) ± √((-100)^2 - 4(6)(405))) / (2(6))

b = (100 ± √13600) / 12

b ≈ 26.87 cm (rounded to two decimal places)

Therefore, the length of side b is approximately 26.87 cm.