Question

Explain how to find the length of a leg of a right triangle if you know the length of the hypotenuse and the length of the other leg.

Short answer

The length of the unknown leg is 8.

Step-by-step solution

Understanding the Context

We are given a right triangle with a known hypotenuse and one leg's length. Our task is to find the length of the other leg. This involves using the Pythagorean Theorem, which relates the sides of a right triangle.

Recall 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 legs (\( a \) and \( b \)): \[ c^2 = a^2 + b^2 \] In this exercise, you know \( c \)and one of the legs, say \( a \). We need to find \( b \).

Rearrange the Formula

To find the unknown leg \( b \), rearrange the Pythagorean Theorem to \( b^2 = c^2 - a^2 \). This equation will give us the square of the unknown leg when the values of \( c \)and \( a \) are substituted.

Substitute the Known Values

Insert the known values of the hypotenuse (\( c \)at 10) and the given leg (\( a \)at 6) into the equation \( b^2 = c^2 - a^2 \). Calculating this gives \( b^2 = 10^2 - 6^2 = 100 - 36 = 64 \).

Solve for the Unknown Leg

Take the square root of both sides of the equation \( b^2 = 64 \)to find the length of the unknown leg \( b \). This results in \( b = \sqrt{64} = 8 \). Thus, the length of the unknown leg is 8.

Key concepts

Right Triangle

In geometry, a right triangle is defined as a triangle with one of its angles measuring exactly 90 degrees. This type of triangle is special because it forms the basis for many mathematical concepts and formulas, such as the Pythagorean Theorem.
A right triangle consists of three sides:
- **Hypotenuse**: This is the longest side and is always opposite the right angle. - **Legs**: The other two sides, which form the right angle.
Understanding a right triangle is crucial in solving numerous geometry problems where angles and side lengths are important. The right-angle (90 degrees) provides a foundation for applying the Pythagorean Theorem and other trigonometric functions that help determine unknown side lengths or angles.

Hypotenuse

The hypotenuse is the longest side of a right triangle, always lying opposite the right angle. Its length is a key element in the Pythagorean Theorem, a fundamental principle in geometry used to relate the sides of right triangles.
The Pythagorean Theorem formula:- Shows that the square of the length of the hypotenuse is equal to the sum of the squares of the lengths of the other two sides.- The formula is written as: \[ c^2 = a^2 + b^2 \]Where \( c \) is the hypotenuse, and \( a \) and \( b \) are the legs.
Knowing the hypotenuse and one leg, you can find the length of the other leg by manipulating and solving the theorem. This method is often used in practical applications such as construction and navigation, where precise measurements are necessary.

Geometry

Geometry, one of the oldest branches of mathematics, is concerned with the properties and relationships of points, lines, surfaces, and solids. In the study of geometry, understanding shapes like right triangles is essential. - Right triangles play an integral part in various geometric concepts, from basic shape properties to more complex theorems and calculations. - The Pythagorean Theorem is a key geometric principle that applies specifically to right triangles, illustrating how geometry helps to determine unknown lengths using known measurements.
Geometry doesn't just involve theoretical applications but also practical ones, such as designing buildings, creating art, and developing technology. Learning geometry helps people understand spatial relationships and provide solutions based on precise calculations.