Problem 144 - Algebra and Trigonometry

Question

An isosceles right triangle has legs that are the same length and acute angles each measuring \(45^{\circ} .\) a. Write an expression in terms of \(a\) that represents the length of the hypotenuse.\ b. Use your result from part (a) to write a sentence that describes the length of the hypotenuse of an isosceles right triangle in terms of the length of a leg.

Step-by-Step Solution

Answer

The length of the hypotenuse of an isosceles right triangle with leg lengths of \(a\) is \(a\sqrt{2}\).

Step 1: Understand the properties of an isosceles right triangle

An isosceles right triangle is a triangle with a right angle (90 degrees) and two 45 degrees angles. Moreover, the sides opposite the equal angles (the legs) are equal.

Step 2: Apply Pythagorean theorem

The Pythagorean theorem states that in a right triangle, the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides. Let's denote the equal legs of the triangle as \(a\). Then, according to Pythagorean theorem, we have \((a)^2 + (a)^2 = (hypotenuse)^2\).

Step 3: Solve the equation for hypotenuse

Solving the equation for the hypotenuse gives \((hypotenuse)^2 = 2(a)^2\), which further simplifies to \(hypotenuse = \sqrt{2(a)^2}= a\sqrt{2}\).

Step 4: Formulate the sentence

Using the derived expression for the hypotenuse, we can form the sentence that describes the length of the hypotenuse of an isosceles right triangle in terms of the length of a leg.

Key Concepts

Pythagorean theorem

The Pythagorean theorem is a fundamental principle in geometry, especially when dealing with right triangles. It provides a way to relate the lengths of the three sides of a right triangle. The theorem is expressed with the equation: \[ c^2 = a^2 + b^2 \]where \(c\) represents the hypotenuse (the side opposite the right angle), and \(a\) and \(b\) are the lengths of the other two sides.

In the context of an isosceles right triangle, this theorem becomes particularly interesting because both of the legs are equal. Thus, it simplifies our work. Since both legs, in this case, are equal, we can use a single variable \(a\) to denote their length.

• This turns our equation into: \((a)^2 + (a)^2 = (hypotenuse)^2\).

The Pythagorean theorem is not only an essential part of solving problems like these but also serves as a gateway to understanding more complex geometrical concepts.

hypotenuse calculation

Calculating the length of the hypotenuse in an isosceles right triangle is straightforward once you apply the Pythagorean theorem. Start by recognizing that since both legs are equal, you only need to consider one variable, which simplifies things.

Using the equation from earlier, \( (hypotenuse)^2 = 2(a)^2 \), we need to find the actual length of the hypotenuse. To do this, take the square root of both sides of the equation to solve for the hypotenuse:

• \( (hypotenuse) = \sqrt{2(a)^2} \)

This simplifies further to:

• \( hypotenuse = a\sqrt{2} \)

Understanding the meaning of the formula \( a\sqrt{2} \) is crucial. It indicates that the hypotenuse is \( \sqrt{2} \) times longer than each of the triangle's legs. This relationship holds true for all isosceles right triangles.

geometry properties

An isosceles right triangle has some unique geometric properties that make it intriguing to study. First, let's break down the triangle's angles and sides.

In any isosceles right triangle:

• The angles measure \(90^{\circ}\), \(45^{\circ}\), and \(45^{\circ}\).
• The two legs opposite the \(45^{\circ}\) angles are equal in length.
• The hypotenuse is always the longest side.

These properties lead to symmetrical triangles, which are often easier to work with because they reduce the complexity in calculations.

Furthermore, when you have an isosceles right triangle, remember that the unique relationship \( hypotenuse = a\sqrt{2} \) emerges from its symmetrical nature. The consistency across all isosceles right triangles reflects the beauty and power of geometry in explaining spatial relationships clearly and predictably.