Question

Heron's formula, \(A=\sqrt{s(s-a)(s-b)(s-c)},\) relates the area, \(A\), of a triangle to the lengths of the three sides, \(a, b,\) and \(c,\) and its semi-perimeter (half its perimeter), \(s=\frac{a+b+c}{2} .\) A triangle has an area of \(900 \mathrm{cm}^{2}\) and one side that measures \(60 \mathrm{cm} .\) The other two side lengths are unknown, but one is twice the length of the other. What are the lengths of the three sides of the triangle?

Short answer

Using Heron's formula and solving polynomial equations, the lengths of the triangle's sides are 60 cm, 45 cm, and 90 cm.

Step-by-step solution

Define known variables

Given the area (A) is 900 cm^2 and one side (a) is 60 cm. The other two sides, b and c, are unknown but one is twice the length of the other. Let b be the unknown side and c = 2b.

Express the semi-perimeter

The semi-perimeter (s) is given by \(s = \frac{a + b + c}{2}\). Substituting a, b, and c gives: \(s = \frac{60 + b + 2b}{2} = \frac{60 + 3b}{2}\).

Apply Heron's formula

Use Heron's formula for the area of the triangle: \(A = \sqrt{s(s-a)(s-b)(s-c)}\). Substituting the known values: \(900 = \sqrt{\frac{60 + 3b}{2} (\frac{60 + 3b}{2} - 60)(\frac{60 + 3b}{2} - b)(\frac{60 + 3b}{2} - 2b)}\).

Simplify and solve for b

Simplify the equation inside the square root. \(900 = \sqrt{\frac{60 + 3b}{2} (\frac{60 + 3b - 120}{2})(\frac{60 + 3b - 2b}{2})(\frac{60 + 3b - 4b}{2})}\). Simplifying further, get the polynomial equation and solve for b.

Solve polynomial equation

Solve for b by simplifying: \(900^2 = \frac{60 + 3b}{2} \times \frac{-60 + 3b}{2} \times \frac{60 + b}{2} \times \frac{60 - b}{2}\). Simplify this step-by-step to find the value of b.

Find side lengths

Evaluate b to find the solutions. Once b is found, compute c = 2b. Verify that the sum of a, b, and cequals the perimeter.

Key concepts

triangle area calculation

Calculating the area of a triangle is a fundamental concept in geometry. One of the most versatile methods for finding the area, especially when side lengths are known, is using Heron's formula. Heron's formula helps calculate the area of any triangle when you know the lengths of all three sides. This is extremely useful in situations where you don’t have the height readily available. For a triangle with sides of lengths a, b, and c, the area, A, can be found using:

Heron's formula: \[A = \sqrt{s(s-a)(s-b)(s-c)}\] Where:

• A is the area of the triangle
• a, b, and c are the lengths of the sides
• s is the semi-perimeter of the triangle

By breaking down the steps and substituting the known values, we can systematically solve for the unknown side lengths and verify our solution using this formula.

Understanding how to manipulate such formulas is key to tackling more complex geometric problems.

algebraic equations

Solving problems using Heron's formula often involves breaking down the problem into algebraic equations. Algebraic manipulation is crucial here as it allows us to simplify and solve for the unknowns.

In our problem, we have one known side, with the other two sides in a specific relationship: one side is twice as long as the other. Let’s take the known variables and set up the relationships:

• If one side is 60 cm (a), and we let b represent the unknown side, then the other side, c, can be expressed as 2b.
• Plugging these into the semi-perimeter formula gives us an expression for s: \[s = \frac{a + b + c}{2} = \frac{60 + b + 2b}{2} = \frac{60 + 3b}{2}\]

Now we substitute s back into Heron's formula and simplify the equation for b:

• \[900 = \sqrt{\frac{60 + 3b}{2} (\frac{60 + 3b}{2} - 60)(\frac{60 + 3b}{2} - b)(\frac{60 + 3b}{2} - 2b)}\]
• By carefully factoring and simplifying, we eventually solve for the value of b.

Mastering the steps of setting up and solving algebraic equations is essential not just for this problem, but for many aspects of math.

semi-perimeter

The semi-perimeter of a triangle is half of its perimeter. It plays an integral role in Heron's formula by serving as part of the input to the equation. Calculating the semi-perimeter is straightforward but crucial:

• The perimeter (P) of a triangle is the sum of its three sides: P = a + b + c.
• The semi-perimeter (s) is then half of this sum: \[s = \frac{a + b + c}{2}\]

In our problem, knowing one side and expressing the other two sides in terms of a single variable allows us to find s easily:

• With a = 60 cm, b as the unknown side, and c = 2b, we have:
• \[s = \frac{60 + b + 2b}{2} = \frac{60 + 3b}{2}\]

Having the semi-perimeter, we then plug it back into Heron's formula to solve for the unknowns. Understanding and working with the semi-perimeter simplifies complex triangular problems considerably.

Knowledge of these fundamental geometric concepts ensures that students can confidently approach and solve various triangle-related problems.