Problem 152 - Elementary Algebra

Question

The hypotenuse of a right triangle is \(10 \mathrm{~cm}\) long. One of the triangle's legs is three times the length of the other leg. Find the lengths of the three sides of the triangle. Round to the nearest tenth.

Step-by-Step Solution

Answer

The lengths of the sides are approximately 3.2 cm, 9.6 cm, and 10.0 cm.

Step 1 - Identify Variables

Let the shorter leg of the triangle be represented by the variable x. The longer leg, being three times the length of the shorter leg, can be represented as 3x. The hypotenuse is given as 10 cm.

Step 2 - Apply the Pythagorean Theorem

The Pythagorean Theorem states that in a right triangle, the square of the hypotenuse (c) is equal to the sum of the squares of the other two sides (a and b). Therefore: \[ a^2 + b^2 = c^2 \] Substitute the known values: \[ x^2 + (3x)^2 = 10^2 \]

Step 3 - Simplify the Equation

Expand and simplify the equation: \[ x^2 + 9x^2 = 100 \] Combine like terms: \[ 10x^2 = 100 \]

Step 4 - Solve for x

Solve for x by dividing both sides of the equation by 10: \[ x^2 = 10 \] Then take the square root of both sides: \[ x = \sqrt{10} \approx 3.2 \text{cm} \]

Step 5 - Find the Lengths of the Legs

Now find the length of the other leg: \[ 3x = 3 \times 3.2 = 9.6 \text{cm} \]

Step 6 - Verify the Solution

Verify that the lengths satisfy the Pythagorean Theorem: \[ (3.2)^2 + (9.6)^2 = 10^2 \] Calculate: \[ 10.24 + 92.16 = 102.4 \] Since 102.4 is close to 100 (considering rounding), the solution is approximately correct.

Key Concepts

Pythagorean Theorem

The Pythagorean Theorem is fundamental in geometry, especially for right triangles. It states that in a right-angled triangle, the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides. This can be expressed as: \[ a^2 + b^2 = c^2 \]Here, \(a\) and \(b\) are the lengths of the legs (shorter sides), and \(c\) is the length of the hypotenuse. This theorem allows you to find one side's length if you know the other two. It's essential for solving problems involving right triangles in various fields such as architecture, physics, and engineering. For example, in our problem, the hypotenuse is 10 cm long. We used the theorem to set up the equation: \[ x^2 + (3x)^2 = 10^2 \] This forms the basis for finding the lengths of the other sides.

Solving Algebraic Equations

Solving algebraic equations often involves isolating the variable. In our problem, we identify the variables and form an equation based on the Pythagorean Theorem. The shorter leg is \(x\), and the longer leg is \(3x\). The equation derived is: \[ x^2 + 9x^2 = 100 \]First, we combine like terms: \[ 10x^2 = 100 \] Next, we isolate \(x\) by dividing both sides by 10: \[ x^2 = 10 \] Finally, we find \(x\) by taking the square root of both sides: \[ x = \sqrt{10} \approx 3.2 \] This process involves straightforward steps in algebra, such as combining like terms and taking square roots. It helps students practice simplifying and solving quadratic equations.

Right Triangle Properties

Understanding the properties of right triangles is crucial. They have a right angle (90 degrees) and two other angles that sum to 90 degrees. Key properties include:

• The relationship between the sides and angles defined by the Pythagorean Theorem.
• The hypotenuse is always opposite the right angle and is the longest side.
• Pythagorean triples, like (3, 4, 5), where all sides are whole numbers.

In our example, the legs are x and 3x. Using properties, we see: \[ x = 3.2 \text{ cm} \] \[ 3x = 9.6 \text{ cm} \] Verifying, we check: \[ 3.2^2 + 9.6^2 = 10^2 \] \[ 10.24 + 92.16 = 102.4 \approx 100 \] So, the calculated side lengths are verified by the properties of a right triangle.