Problem 17 - Algebra 2 and Trigonometry

Question

The length of one of the equal sides of an isosceles triangle measures 25.8 inches and each base angle measures 53 degrees. a. Find the measure of the base of the triangle to the nearest tenth. b. Find the perimeter of the triangle to the nearest inch.

Step-by-Step Solution

Answer

The base is approximately 31.1 inches, and the perimeter is 83 inches.

Step 1: Understand the Triangle

The given triangle is isosceles with equal sides measuring 25.8 inches each and base angles of 53 degrees. This means the two base angles are equal, and the triangle has two equal sides and one base.

Step 2: Identify Known Measurements

We know two sides of the triangle (both 25.8 inches) and two base angles (each 53 degrees). We need to find the length of the base and the perimeter.

Step 3: Use the Law of Cosines

To find the base length of the triangle, we can use the Law of Cosines: \[ c^2 = a^2 + b^2 - 2ab \cdot \cos(C) \]Where \( a \) and \( b \) are the equal sides, \( c \) is the base, and \( C \) is the angle opposite the base (which is \( 180 - 2 \times 53 = 74 \) degrees).

Step 4: Plug in the Known Values

Using the Law of Cosines formula, we substitute the values:\[ c^2 = 25.8^2 + 25.8^2 - 2 \cdot 25.8 \cdot 25.8 \cdot \cos(74°) \]

Step 5: Calculate the Base Length

Calculate the cosine and simplify the equation:\[ c^2 = 666.64 + 666.64 - 1333.28 \cdot 0.2756 \] \[ c^2 = 666.64 + 666.64 - 367.17 \] \[ c^2 = 966.11 \] \[ c = \sqrt{966.11} \approx 31.1 \]The base length is approximately 31.1 inches, rounded to the nearest tenth.

Step 6: Calculate the Perimeter of the Triangle

To find the perimeter, add up all sides:\[ \text{Perimeter} = 25.8 + 25.8 + 31.1 \approx 82.7 \]Rounded to the nearest inch, the perimeter is 83 inches.

Key Concepts

law of cosines

To solve for unknown sides in a triangle, especially when dealing with non-right triangles, the Law of Cosines is a handy tool. This formula helps when we know:

• Two sides of a triangle.
• The angle between those two sides.

In our isosceles triangle, we applied the Law of Cosines to determine the missing base length. The formula is given by:\[ c^2 = a^2 + b^2 - 2ab \cdot \cos(C) \]- Here, \(a\) and \(b\) represent the known equal sides of the isosceles triangle.- \(C\) represents the angle opposite the base.- \(c\) will be our unknown base, which we solve for. By substituting the known values, we were able to solve for \(c\) using arithmetic. The Law of Cosines effectively modified our basic understanding of quadratic equations by including trigonometric elements to handle angles and sides together.

base angles

In an isosceles triangle, the two angles opposite the equal sides are known as the base angles. In every isosceles triangle:

• The base angles are equal.
• Knowing the angle information helps determine the third angle by utilizing the angle sum property of triangles, which states that the sum of all angles in a triangle is always 180°.

For our exercise, the base angles each measure 53 degrees. This enabled us to find the third angle, which is opposite to the base:\[ 180 - (2 \times 53) = 74 \]\degreesKnowing all the angles facilitated the use of the Law of Cosines, ensuring we correctly identified the angle being used in cosine calculations. Base angles not only help in finding unknown angles but also are essential in setting up equations used in solving side lengths of triangles.

triangle perimeter

The perimeter of a triangle is the total length of its boundaries. For isosceles triangles, calculating the perimeter becomes straightforward once we know:

• The length of the two equal sides.
• The length of the base.

Once the length of the base (31.1 inches) was calculated using the Law of Cosines, adding it to the lengths of the equal sides allowed us to find the perimeter:\[ \text{Perimeter} = 25.8 + 25.8 + 31.1 \]Simplifying gives:\[ \text{Perimeter} \approx 82.7 \] in inches.Finally, rounding to the nearest inch, we found the perimeter to be 83 inches. Perimeter calculations are crucial in many real-world applications, such as in gear ratios, material cost estimation, and for architectural designs in ensuring accurate measurements.