Problem 21 - 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. \(a=10 \mathrm{cm}\) and \(c=26 \mathrm{cm}\) (RIGHT TRIANGLE CAN'T COPY)

Step-by-Step Solution

Answer

The length of the missing side is 24 cm.

Step 1: Identify the given sides

We are given the lengths of side \( a = 10 \text{ cm} \) and the hypotenuse \( c = 26 \text{ cm} \) of the right triangle \( \triangle ABC \). We need to find the length of side \( b \).

Step 2: Apply the Pythagorean Theorem

In a right triangle, the Pythagorean Theorem states: \[ a^2 + b^2 = c^2 \]Substitute the known values: \[ 10^2 + b^2 = 26^2 \]

Step 3: Calculate squares of known sides

Calculate the square of side \( a \) and the hypotenuse \( c \): \[ 10^2 = 100 \] and \[ 26^2 = 676 \]

Step 4: Solve for the unknown side

Substitute the calculated squares into the equation: \[ 100 + b^2 = 676 \]Subtract 100 from both sides to isolate \( b^2 \): \[ b^2 = 576 \]

Step 5: Find the length of the missing side

Take the square root of both sides to solve for \( b \): \[ b = \sqrt{576} = 24 \text{ cm} \]

Key Concepts

Understanding Right Triangles

A right triangle is unique because it has one angle exactly equal to 90 degrees. This specific angle helps define other properties of the triangle, making it a crucial concept in geometry. In a right triangle, the side opposite this right angle is known as the hypotenuse, which is also the longest side of the triangle.
When working with right triangles, it's important to recognize the relationship between the sides. The two sides that form the right angle are typically known as the legs of the triangle. In mathematical problems, like the one given where side lengths are involved, understanding the arrangement and names of these sides helps set the ground for applying critical formulas such as the Pythagorean Theorem.

• The right angle is key to defining a right triangle.
• The hypotenuse is opposite the right angle and is the longest side.
• The other two sides are referred to as legs.

What is the Hypotenuse?

The hypotenuse plays a central role in right triangles. It is the side opposite the right angle and due to its geometric position, it is always the longest side of the triangle. This is because, in a right triangle, it directly connects the points of the two other sides forming the right angle. In our problem, the hypotenuse is already given as 26 cm. Having this side length allows us to use the Pythagorean Theorem to find the missing side length. Remember, the hypotenuse is always used with its square value in the theorem. Conclusively:

• The hypotenuse is opposite the 90-degree angle.
• It is the longest side of a right triangle.
• In calculations, its squared value is a part of the Pythagorean relation.

Pythagorean Theorem in Geometry

The Pythagorean Theorem is a fundamental principle in geometry, especially when dealing with right triangles. Formulated as \(a^2 + b^2 = c^2\), it relates the lengths of the sides of a right triangle. The letters \(a\) and \(b\) represent the legs, while \(c\) denotes the hypotenuse.In practical applications, such as our exercise, the theorem helps find a missing side length if the other two are known. Steps to apply the theorem include:

• Identify the known side lengths.
• Substitute these values into the theorem's formula.
• Solve the equation for the unknown side using algebraic techniques, such as isolating the variable or taking square roots.

With these steps, understanding and solving for side lengths in right triangles becomes manageable. For any right triangle, the Pythagorean Theorem remains a powerful tool linking algebra and geometry.