Question

If a right triangle has hypotenuse \(c=2\) and leg \(a=1,\) find the length of the other leg \(b\).

Short answer

The length of the other leg b is sqrt{3} .

Step-by-step solution

Understand the Pythagorean Theorem

For any right triangle, the Pythagorean Theorem states that the sum of the squares of the two legs is equal to the square of the hypotenuse. Mathematically, this is written as: a^2 + b^2 = c^2.

Substitute Given Values

Plug in the known values into the Pythagorean Theorem. Here, we know that the hypotenuse ( c ) is 2 and one of the legs ( a ) is 1. So, the equation becomes: 1^2 + b^2 = 2^2.

Simplify the Equation

Simplify the equation by performing the squaring operations: 1 + b^2 = 4.

Isolate the Variable

Isolate b^2 by subtracting 1 from both sides of the equation: b^2 = 4 - 1 b^2 = 3.

Solve for b

Take the square root of both sides of the equation to solve for b : b = sqrt{3} .

Key concepts

Right Triangle

A right triangle is a type of triangle that has one angle which is exactly 90 degrees. This means that it forms a perfect 'L' shape, which is useful in many mathematical applications.
Right triangles have three sides: two legs and the hypotenuse. The legs are the two sides that form the right angle. The longest side, opposite the right angle, is called the hypotenuse. Right triangles are crucial in various fields, including architecture, engineering, and physics.
Understanding right triangles helps in solving many geometric problems, especially those involving distances and heights. It's also the foundation for trigonometry.

Hypotenuse

In any right triangle, the hypotenuse is the side opposite the right angle and is always the longest side.
To identify the hypotenuse, look for the side that does not form the right angle. This side spans across the triangle, creating a straight line between the two legs.
In our problem, the hypotenuse is given as 2. This information is essential because it helps us apply the Pythagorean Theorem correctly. By knowing the hypotenuse and one leg, we can find the missing length of the other leg using the theorem.

Solving Equations

Solving equations involves finding the value of an unknown variable that makes the equation true. This process is a key skill in mathematics and applies to our exercise.
We start with the Pythagorean Theorem: ewline ewline ewline ewline a^2 + b^2 = c^2.
By substituting the given values of the hypotenuse (c) and one leg (a), we form a new equation: ewline ewline 1^2 + b^2 = 2^2.
Next, simplify and isolate the variable. Perform the operations to make it easier to solve: ewline ewline 1 + b^2 = 4 -> b^2 = 4 - 1 -> b^2 = 3.
Finally, we solve for b. We will explain this in the next section.

Square Root

The square root is the value that, when multiplied by itself, gives the original number. It is denoted by the symbol \ \( \sqrt{}...\ \) \.
To find the length of the other leg (b) in our problem, we solve \( b^2 = 3 \).
Taking the square root of both sides gives: \[ b = \ \sqrt{}3 \].
This means the length of the other leg in the right triangle is \( \ \sqrt{}3 \).
Understanding square roots is vital for solving equations involving quadratic terms, often seen not just in geometry but also in algebra and physics.