Question

example 7

1. geometry heron’s formula states that the area of a triangle whose sides have lengths (a), (b), and (c) is (a = sqrt{s(s - a)(s - b)(s - c)}) where (s = \frac{1}{2}(a + b + c)).

if the area of the triangle is (270space cm^2), (s = 45space cm), (a = 15space cm), and (c = 39space cm), what is the length of side (b)?

Explanation:

Step1: Substitute known values into Heron's formula

We know \( A = 270 \), \( s = 45 \), \( a = 15 \), \( c = 39 \), and \( A=\sqrt{s(s - a)(s - b)(s - c)} \). Substituting the values, we get \( 270=\sqrt{45(45 - 15)(45 - b)(45 - 39)} \).

Step2: Simplify the equation

First, simplify the terms inside the square root: \( 45-15 = 30 \), \( 45 - 39=6 \). So the equation becomes \( 270=\sqrt{45\times30\times(45 - b)\times6} \). Calculate \( 45\times30\times6=45\times180 = 8100 \). Now the equation is \( 270=\sqrt{8100\times(45 - b)} \).

Step3: Square both sides of the equation

Squaring both sides, we get \( 270^{2}=8100\times(45 - b) \). Calculate \( 270^{2}=72900 \). So \( 72900 = 8100\times(45 - b) \).

Step4: Solve for \( (45 - b) \)

Divide both sides by 8100: \( \frac{72900}{8100}=45 - b \). \( \frac{72900}{8100}=9 \), so \( 9 = 45 - b \).

Step5: Solve for \( b \)

Rearrange the equation: \( b=45 - 9 \). So \( b = 36 \).

Answer:

The length of side \( b \) is \( \boldsymbol{36} \) cm.