Problem 10 - Prealgebra

Question

Your solutions should include a well-labeled sketch. The length of one leg of a right triangle is 5 meters, and the length of the hypotenuse is 10 meters. Find the exact length of the other leg.

Step-by-Step Solution

Answer

The length of the other leg is \(5\sqrt{3}\) meters.

Step 1: Understand the Problem

In a right triangle, one leg is known to be 5 meters, and the hypotenuse is 10 meters. We need to find the length of the other leg using the Pythagorean theorem.

Step 2: Set Up the Pythagorean Theorem

The Pythagorean theorem states that for a right triangle, the sum of the squares of the lengths of the legs equals the square of the length of the hypotenuse. This is expressed as:\[ a^2 + b^2 = c^2 \]where \(a\) and \(b\) are the legs, and \(c\) is the hypotenuse. Here, \(a = 5\) meters and \(c = 10\) meters.

Step 3: Substitute Known Values

Substitute the known values into the equation:\[ 5^2 + b^2 = 10^2 \]

Step 4: Perform Calculations

Calculate the squares of the known values:\[ 25 + b^2 = 100 \]Subtract 25 from both sides to solve for \(b^2\):\[ b^2 = 75 \]

Step 5: Solve for the Missing Leg

To find \(b\), take the square root of both sides:\[ b = \sqrt{75} \]Simplifying \(\sqrt{75}\), we find:\[ b = \sqrt{25 \times 3} = 5\sqrt{3} \] meters.

Step 6: Sketch the Right Triangle

Sketch a right triangle labeling one leg as 5 meters, the hypotenuse as 10 meters, and the other leg as \(5\sqrt{3}\) meters. This sketch visually confirms the calculated values.

Key Concepts

Pythagorean theorem

The Pythagorean theorem is a fundamental principle in geometry dealing with right triangles. It states that in a right 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 lengths of the other two sides. This can be expressed mathematically as:\[ a^2 + b^2 = c^2 \]where \(a\) and \(b\) are the legs of the triangle, and \(c\) is the hypotenuse. This relationship is crucial for solving triangles, especially to find an unknown side when the other two are known. By mastering the Pythagorean theorem, students can tackle a wide variety of geometry problems and deepen their understanding of triangle properties.

solving triangles

Solving triangles involves finding missing sides or angles when certain elements of the triangle are known. In the context of a right triangle, like the one given in our exercise, solving involves using the Pythagorean theorem to find an unknown side.For instance, if you know one leg and the hypotenuse, you can find the other leg. To do this:

• Write down the Pythagorean theorem: \( a^2 + b^2 = c^2 \)
• Substitute the known side lengths.
• Perform the arithmetic to solve for the unknown side.

Once you find the unknown side, it's good practice to sketch the triangle. This visual representation helps verify the accuracy of your calculations and provides a clear understanding of the triangle's dimensions.

square roots

The concept of square roots is essential in solving geometric problems involving the Pythagorean theorem. A square root is a value that, when multiplied by itself, gives the original number. For instance, the square root of 25 is 5, because \(5 \times 5 = 25\).When solving for a missing side in a right triangle, as in our exercise, you may end up with an equation like \(b^2 = 75\). To find \(b\), take the square root of both sides, giving you \(b = \sqrt{75}\). This process is crucial since it translates squared lengths back into their linear dimensions.Square roots can sometimes be simplified. In our example, \(\sqrt{75} = \sqrt{25 \times 3} = 5\sqrt{3}\). Recognizing and simplifying square roots is an important skill. It helps present the answer in its most comprehensible form, which is particularly useful in mathematical or geometric contexts.