Problem 4 - Algebra 1

Question

Find the missing length of the right triangle if a and b are the lengths of the legs and c is the length of the hypotenuse. $$a=7, b=24$$

Step-by-Step Solution

Answer

The missing length of the right triangle's hypotenuse, \(c\), is 25.

Step 1: State the Pythagorean theorem and insert known values

The Pythagorean theorem is: \(a^2 + b^2 = c^2\). Substitute the known values for \(a\) and \(b\) into this equation: \(7^2 + 24^2 = c^2\).

Step 2: Perform the calculations

Calculate the squares of 7 and 24 and sum them, which equals the square of the hypotenuse \(c\). So \(49 + 576 = c^2\), which simplifies to \(625 = c^2\).

Step 3: Solve for the hypotenuse length

To find \(c\), we need to take the square root of both sides of the equation: \(\sqrt{625} = c\).

Key Concepts

Right Triangle

A right triangle is a special type of triangle where one angle is exactly 90 degrees, often referred to as a right angle. The sides forming the right angle are called the 'legs,' while the side opposite the right angle is known as the 'hypotenuse.' In a right triangle, the hypotenuse is always the longest side. This is a consequence of the Pythagorean theorem, which establishes a specific relationship between the lengths of the sides of a right triangle.

In any right triangle, the square of the hypotenuse length is equal to the sum of the squares of the other two sides, a fundamental property that allows us to solve for unknown sides in these triangles. Understanding this concept sets the basis for addressing complex geometric problems and also has real-world applications, such as in navigation, construction, and even art.

Hypotenuse Length

The hypotenuse length in a right triangle can be found using the Pythagorean theorem. This theorem provides a straightforward method for calculating the length of the hypotenuse when the lengths of the other two sides (legs) are known. In essence, if the lengths of the legs are referred to as 'a' and 'b', and the hypotenuse as 'c', then the theorem is represented by the formula:
\[a^2 + b^2 = c^2\].

Applying this formula, we can solve for the hypotenuse length by rearranging it to \(c = \sqrt{a^2 + b^2}\). For instance, if we have a right triangle with legs of 7 and 24 units, we plug these values into our formula, resulting in \(c = \sqrt{7^2 + 24^2}\) and finally calculating \(c = \sqrt{49 + 576}\), which equals \(c = \sqrt{625}\). This process illustrates how we derive the hypotenuse length from the two shorter sides of the triangle.

Squares and Square Roots

Squares and square roots are mathematical concepts that play a crucial role in understanding the Pythagorean theorem. The 'square' of a number is that number multiplied by itself, represented as \(x^2 = x \times x\). For instance, the square of 7 is 49 because \(7^2 = 7 \times 7\).

The 'square root' is the reverse operation. It is designated by the symbol \(\sqrt{}\) and represents what number, when squared, equals the value under the square root. For instance, \(\sqrt{49} = 7\) because 7 is the number that, when multiplied by itself, equals 49. In the context of the Pythagorean theorem, after finding the square of the hypotenuse, we use the square root to return to the original value of the hypotenuse length. So, using our example, we found the hypotenuse to be \(c^2 = 625\) by squaring the lengths of the legs. To find 'c', we take the square root of 625, which is 25, and therefore, the hypotenuse length is 25 units.