Problem 44 - Prealgebra

Question

The hypotenuse of an isosceles right triangle is 8 inches. Is there enough information to find the length of the legs? If so, find the length of the legs. If not, explain why not.

Step-by-Step Solution

Answer

Yes, the length of each leg is \( 4\sqrt{2} \) inches.

Step 1: Understand Isosceles Right Triangle

An isosceles right triangle is a triangle with two equal sides and one right angle. In this case, the two equal sides are the legs of the triangle, and the hypotenuse is the side opposite the right angle.

Step 2: Identify Relationship for Right Triangle

In a right-angled triangle, the relationship between the hypotenuse and the legs is given by the Pythagorean theorem: \[ c^2 = a^2 + b^2 \], where \( c \) is the hypotenuse and \( a \) and \( b \) are the legs. For an isosceles right triangle, \( a = b \).

Step 3: Express Hypotenuse in Terms of Legs

Since the triangle is isosceles, we use the formula for its hypotenuse: \( c = a\sqrt{2} \). Given that the hypotenuse is 8 inches, we set up the equation: \[ 8 = a\sqrt{2} \].

Step 4: Solve for Leg Length

Rearrange the equation to solve for \( a \): \[ a = \frac{8}{\sqrt{2}} \]. Rationalize the denominator, which gives: \[ a = \frac{8\sqrt{2}}{2} = 4\sqrt{2} \]. Therefore, the length of each leg is \( 4\sqrt{2} \) inches.

Key Concepts

Pythagorean Theorem

The Pythagorean Theorem is a fundamental principle in geometry. It applies specifically to right-angled triangles. The theorem states that in such triangles, the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the lengths of the other two sides. This is mathematically represented as \[c^2 = a^2 + b^2\] where \(c\) represents the hypotenuse, and \(a\) and \(b\) stand for the two legs.
In the case of an isosceles right triangle, the situation is even simpler. Since both legs are of equal length, we denote both as \(a\). Hence, the equation simplifies to \[c^2 = a^2 + a^2 = 2a^2\] This equation is particularly useful in determining unknown side lengths when given enough initial information about the triangle.

Hypotenuse-Leg Relationship

In an isosceles right triangle, the relationship between the hypotenuse and the legs is unique. Since the angles are \(45^\circ\), \(45^\circ\), and \(90^\circ\), the hypotenuse can always be described in terms of the legs.
For any isosceles right triangle, the hypotenuse \(c\) is given by the formula: \[c = a\sqrt{2}\] where \(a\) is the length of each leg.
This formula stems from the property of these triangles where each leg is equal, creating a predictable pattern in relation to the hypotenuse. Determining the length of the hypotenuse or the legs becomes straightforward when one component is known.

• This provides a direct relationship to quickly evaluate side lengths without recalculating everything from scratch each time.
• It highlights the symmetry in these triangles.

Rationalization in Algebra

Rationalization is an algebraic technique used to eliminate radicals from the denominator of a fraction. It makes the expression simpler and easier to understand or compare.
In the problem at hand, to find the length of the legs of an isosceles right triangle with a hypotenuse of 8 inches, we start by setting up the equation: \[a = \frac{8}{\sqrt{2}}\] Since having a square root in the denominator is not ideal, we rationalize it by multiplying both the numerator and the denominator by the square root present in the denominator, \(\sqrt{2}\): \[a = \frac{8}{\sqrt{2}} \times \frac{\sqrt{2}}{\sqrt{2}} = \frac{8\sqrt{2}}{2} = 4\sqrt{2}\] Now, the expression \(4\sqrt{2}\) is much cleaner, with no radical in the denominator, making it easier to handle in subsequent calculations.

• This step ensures clarity and precision in mathematical solutions.
• Rationalizing is especially useful in exams or when presenting results clearly.