Problem 48 - Algebra 2 and Trigonometry

Question

The perimeter of an isosceles triangle is \(\sqrt{50}\) feet. The lengths of the sides are in the ratio \(3 : 3 : 4\) . Find the length of each side of the triangle.

Step-by-Step Solution

Answer

The side lengths are \(\frac{3\sqrt{2}}{2}\) feet, \(\frac{3\sqrt{2}}{2}\) feet, and \(2\sqrt{2}\) feet.

Step 1: Identify the Variables

Let the length of the equal sides of the isosceles triangle be represented as \(3x\) each and the base be represented as \(4x\). These come from the given side ratio of 3:3:4.

Step 2: Set Up the Equation for Perimeter

The perimeter \(P\) of the triangle is given by the sum of its sides. Hence: \[ P = 3x + 3x + 4x = 10x \] We know that the perimeter \(P\) is \(\sqrt{50}\) feet. So, we can set:\[ 10x = \sqrt{50} \]

Step 3: Solve for \(x\)

To find \(x\), divide both sides of the equation by 10:\[ x = \frac{\sqrt{50}}{10} \]Simplify \(\sqrt{50}\) as \(5\sqrt{2}\), and the equation becomes:\[ x = \frac{5\sqrt{2}}{10} = \frac{\sqrt{2}}{2} \]

Step 4: Find the Length of Each Side

Using \(x = \frac{\sqrt{2}}{2}\), the lengths of the sides are:- The two equal sides: \(3x = 3 \times \frac{\sqrt{2}}{2} = \frac{3\sqrt{2}}{2}\) feet- The base: \(4x = 4 \times \frac{\sqrt{2}}{2} = 2\sqrt{2}\) feet

Key Concepts

Perimeter

The perimeter of any polygon is the total length around it, obtained by summing the lengths of all its sides. For an isosceles triangle, which has two equal sides, the perimeter is found by adding these two equal sides and the base.
For example, if the lengths of the two equal sides of an isosceles triangle are both 3 units, and the base is 4 units, then the perimeter is calculated as follows:

• Perimeter = 3 + 3 + 4
• Perimeter = 10 units

In our specific problem, the perimeter is known to be \(\sqrt{50}\) feet. By setting this equal to the sum of the sides, we can calculate unknown values like the scaling factor (\(x\)) or individual side lengths. Calculating the perimeter in this way helps us determine possible side lengths that fit both the given perimeter and side ratios.

Ratios

Ratios are a way of expressing the relation between quantities. In an isosceles triangle where sides have a ratio of \(3:3:4\), it reflects the comparative lengths of the two equal sides and the base.
For an easier understanding, consider that ratios stay consistent by multiplying a common factor to find actual lengths. If you assume these sides as \(3x, 3x,\) and \(4x\) respectively, the ratio is preserved by the common multiplier \(x\). Here is how it works in our case:

• Each side is multiplied by \(x\)
• This gives sides \(3x, 3x,\) and \(4x\)

By maintaining the ratio 3:3:4, while using the perimeter equation, we solve for \(x\), thereby determining the actual lengths of the sides from the triangle's known perimeter.

Simplifying Radicals

Simplifying radicals involves expressing a square root in its simplest form. This is crucial in making equations manageable and easier to compute, specifically when dealing with irrational numbers.
For example, consider \(\sqrt{50}\). It can be expressed as \(\sqrt{25 \times 2}\), which simplifies to \(5\sqrt{2}\) since \(\sqrt{25} = 5\). This process involves:

• Identifying the largest perfect square factor of the number under the square root
• Splitting the original number into this factor and another number
• Taking the square root of the perfect square separately

In our exercise, simplifying \(\sqrt{50}\) is necessary to solve for \(x\) in the equation \(10x = \sqrt{50}\). This expresses \(x\) in a simpler radical form, further leading to accurate calculations of each triangle side's length.