Question

In a right triangle with legs \(a=6\) and \(b=10\), the Pythagorean Theorem tells us that the hypotenuse is \(c=______\)

Short answer

\( c = 2 \sqrt{34} \)

Step-by-step solution

Identify the Problem

The problem asks to find the hypotenuse of a right triangle with given legs using the Pythagorean Theorem.

Write the Pythagorean Theorem

The Pythagorean Theorem is given by the equation: \[ a^2 + b^2 = c^2 \] where \( a \) and \( b \) are the lengths of the legs, and \( c \) is the length of the hypotenuse.

Substitute the Given Values

Substitute \( a = 6 \) and \( b = 10 \) into the equation: \[ 6^2 + 10^2 = c^2 \]

Calculate the Squares of the Legs

Calculate the squares: \[ 6^2 = 36 \] \[ 10^2 = 100 \]

Add the Results

Add the results: \[ 36 + 100 = 136 \]

Solve for the Hypotenuse

Take the square root of both sides to find \( c \): \[ c = \sqrt{136} \] Simplify if possible: \[ c = \sqrt{4 \times 34} = 2 \sqrt{34} \]

Key concepts

right triangle

A right triangle is a type of triangle that includes one angle measuring 90 degrees. This right angle creates a special relationship between the three sides of the triangle. The two shorter sides are called the legs, and the longest side is called the hypotenuse.

In any right triangle, the legs (let's call them `a` and `b`) and the hypotenuse (`c`) are connected by a famous formula known as the Pythagorean Theorem. This theorem is essential for solving various geometry problems.

For instance, in our problem, we have legs `a = 6` and `b = 10`. By applying the Pythagorean Theorem, we can solve for the hypotenuse.

hypotenuse

The hypotenuse is the longest side of a right triangle. It is the side that is opposite the right angle. Using the Pythagorean Theorem, you can find the hypotenuse if you know the lengths of the other two sides.

The formula for the Pythagorean Theorem is \( a^2 + b^2 = c^2 \). This shows that the square of the hypotenuse (\( c \)) is equal to the sum of the squares of the legs (\( a \) and \( b \)).

In the given problem, we substitute \( a = 6 \) and \( b = 10 \) into the theorem: \[ 6^2 + 10^2 = c^2 \]. This simplifies to \[ 36 + 100 = c^2 \]. Adding these up, we get \[ 136 = c^2 \]. By taking the square root, we find the hypotenuse: \[ c = \sqrt{136} = 2 \sqrt{34} \].

square root

The square root is a mathematical operation that returns a number which, when multiplied by itself, gives the original number. It is denoted by the radical symbol \( \sqrt{} \).

For example, the square root of 25 is 5, because \[ 5 \times 5 = 25 \]. Similarly, to solve for the hypotenuse in our problem, we arrived at the equation \[ c^2 = 136 \]. To isolate the hypotenuse, we take the square root of both sides: \[ c = \sqrt{136} \].

Often, the square root can be simplified if the number under the radical is a product of a perfect square. Here, \sqrt{136} \ can be simplified to \sqrt{4 \times 34} = 2 \sqrt{34} \. Simplifying square roots can make calculations easier and expressions cleaner.