Math Problem Statement

Evaluate the given formula A = sqrt(s(s-a)(s-b)(s-c)) for a = 15 1/2 in., b = 13 1/2 in., c = 12 1/4 in., s = 20 in.

Solution

The given formula represents Heron's formula for the area AA of a triangle when the lengths of its three sides aa, bb, and cc are known, along with the semi-perimeter ss:

A=s(sa)(sb)(sc)A = \sqrt{s(s - a)(s - b)(s - c)}

The problem provides the following values:

  • a=1512a = 15 \frac{1}{2} inches (which is a=15.5a = 15.5 inches),
  • b=1312b = 13 \frac{1}{2} inches (which is b=13.5b = 13.5 inches),
  • c=1214c = 12 \frac{1}{4} inches (which is c=12.25c = 12.25 inches),
  • s=20s = 20 inches.

Step-by-step solution:

  1. Subtract each side from the semi-perimeter:

    • sa=2015.5=4.5s - a = 20 - 15.5 = 4.5
    • sb=2013.5=6.5s - b = 20 - 13.5 = 6.5
    • sc=2012.25=7.75s - c = 20 - 12.25 = 7.75
  2. Substitute all the values into the formula: A=20(4.5)(6.5)(7.75)A = \sqrt{20(4.5)(6.5)(7.75)}

  3. Calculate the product: 20×4.5=9020 \times 4.5 = 90 90×6.5=58590 \times 6.5 = 585 585×7.75=4533.75585 \times 7.75 = 4533.75

  4. Take the square root: A=4533.7567.32square inchesA = \sqrt{4533.75} \approx 67.32 \, \text{square inches}

Thus, the area of the triangle is approximately A67.32square inchesA \approx 67.32 \, \text{square inches}.

Would you like more details on any of the steps, or further clarifications?

Here are 5 related questions for further exploration:

  1. How would the area change if the semi-perimeter were different?
  2. What does the semi-perimeter represent geometrically?
  3. Can Heron’s formula be used for a right triangle, and how does it compare with the usual area formula for such triangles?
  4. What would the area be if the triangle were equilateral, with all sides equal?
  5. How do we handle a situation where the semi-perimeter is smaller than one of the sides?

Tip: When using Heron's formula, always ensure that the sum of any two sides is greater than the third, as required by the triangle inequality.