Problem 20 - Elementary and Intermediate Algebra

Question

The lengths of two sides of the right triangle \(A B C\) are given. Find the length of the missing side. \(b=18 \mathrm{m}\) and \(c=82 \mathrm{m}\) (RIGHT TRIANGLE CAN'T COPY)

Step-by-Step Solution

Answer

The missing side \( a \) is 80 m.

Step 1: Identify the Triangle Sides

Since the triangle is a right triangle and we're given the length of side \( b \) (one leg) and side \( c \) (the hypotenuse), we need to find the length of side \( a \) (the other leg).

Step 2: Recall the Pythagorean Theorem

For a right triangle, the Pythagorean Theorem states that \( a^2 + b^2 = c^2 \). We will use this formula to find the missing side \( a \).

Step 3: Substitute Given Numbers into the Formula

Substitute \( b = 18 \) m and \( c = 82 \) m into the equation. This becomes: \[ a^2 + (18)^2 = (82)^2 \]

Step 4: Calculate the Squares

Calculate \( 18^2 = 324 \) and \( 82^2 = 6724 \). Substitute these values into the equation:\[ a^2 + 324 = 6724 \]

Step 5: Solve for \( a^2 \)

Subtract 324 from both sides of the equation to isolate \( a^2 \):\[ a^2 = 6724 - 324 = 6400 \]

Step 6: Find \( a \)

Take the square root of both sides to find \( a \):\[ a = \sqrt{6400} = 80 \] m.

Key Concepts

Understanding the Right Triangle

A right triangle is a type of triangle that has one angle measuring exactly 90 degrees. This right angle creates a special relationship between the lengths of the sides, which are known as the legs and the hypotenuse.
A typical right triangle consists of:

• Two legs: These are the sides that form the right angle. In our problem, one of these sides is given as 18 meters, labeled as side \( b \).
• Hypotenuse: This is the side opposite the right angle and is always the longest side of the triangle. In our scenario, the hypotenuse is 82 meters, labeled as side \( c \).

In a right triangle, the lengths of these sides are connected through a crucial formula known as the Pythagorean Theorem. This formula is the key to understanding how a right triangle works and is used to solve exercises involving missing side calculations.

Calculating a Missing Side with the Pythagorean Theorem

When faced with the problem of calculating a missing side in a right triangle, the Pythagorean Theorem is our go-to tool. This theorem establishes the relationship between the sides of the triangle, ensuring the missing side can be calculated if the other two sides are known.
The Pythagorean Theorem formula is: \( a^2 + b^2 = c^2 \). Here, \( a \) and \( b \) represent the legs of the triangle, while \( c \) is the hypotenuse. This equation helps determine any missing side if two sides are known.

• Given side \( b = 18 \) m and side \( c = 82 \) m, our task is to find side \( a \). This results in the equation \( a^2 + 18^2 = 82^2 \).
• After calculating \( 18^2 = 324 \) and \( 82^2 = 6724 \), the equation becomes \( a^2 + 324 = 6724 \).
• By subtracting 324 from 6724, we isolate \( a^2 \): \( a^2 = 6400 \).

Finally, to find the length of \( a \), we take the square root of \( 6400 \), arriving at \( a = 80 \) m. This calculation demonstrates the elegance and practicality of the Pythagorean Theorem in solving missing side problems.

Approaching Mathematical Problem-Solving with Confidence

Mathematical problem-solving can feel daunting, but breaking down the process into manageable steps can simplify seemingly complex problems like those involving right triangles.
When tackling such a problem, it's essential to approach it methodically:

• First, identify what is given and what needs to be found. Clearly label the sides of the triangle to avoid confusion.
• Next, utilize established formulas like the Pythagorean Theorem, aligning the known numbers correctly within the equation.
• Proceed with calculations systematically. Solve for any unknowns, always keeping track of units such as meters in this case.
• Finally, review your solution to ensure it makes sense within the context of the problem. Check all calculations to confirm accuracy.

Through practice and consistency, problem-solving becomes more intuitive. Visualizing the steps and verifying each part can boost confidence and proficiency in mathematics, allowing you to tackle a wide range of exercises with assurance.