Problem 7 - Intermediate Algebra

Question

Fill in the blanks. In an isosceles right triangle, the length of the hypotenuse is \(\square\) times the length of one leg.

Step-by-Step Solution

Answer

\(\sqrt{2}\) times the length of one leg.

Step 1: Recognize the Triangle Type

An isosceles right triangle is a triangle with a right angle and two equal sides. In this case, the two equal sides, also known as the legs, are also the equal angles. The angles are both 45°.

Step 2: Pythagorean Theorem Setup

In a right triangle, the Pythagorean theorem relates the lengths of the sides with the equation: \[ c^2 = a^2 + b^2 \]In an isosceles right triangle, both legs \(a\) and \(b\) are of equal length \(x\). The equation simplifies to: \[ c^2 = x^2 + x^2 \]

Step 3: Simplify the Equation

Combine like terms in the equation: \[ c^2 = 2x^2 \]

Step 4: Solve for the Hypotenuse

To find the length of the hypotenuse \(c\), take the square root of both sides: \[ c = \sqrt{2x^2} \]This simplifies to:\[ c = x\sqrt{2} \]

Step 5: Conclusion

The expression \(x\sqrt{2}\) shows that the hypotenuse is \(\sqrt{2}\) times the length of one leg. Therefore, the hypotenuse is \(\sqrt{2}\) times the length of one leg.

Key Concepts

Pythagorean Theorem

The Pythagorean Theorem is a cornerstone in the study of triangle geometry. It provides a relationship between the lengths of the sides of a right triangle. Defined by the equation \( c^2 = a^2 + b^2 \), where \(c\) is the hypotenuse and \(a\) and \(b\) are the other two sides. In an isosceles right triangle, where both the legs are equal, the theorem becomes even more specific: \( c^2 = x^2 + x^2 \). This simplifies the relationship to \( c^2 = 2x^2 \). This shows us directly how the hypotenuse is connected to the legs. Believing in this theorem's reliability allows us to solve many geometric problems when precise side length relationships are crucial.

Hypotenuse

The hypotenuse is the longest side of a right triangle and is always opposite the right angle. In the case of an isosceles right triangle, figuring out the hypotenuse involves a consistent geometric principle.Given both legs are equal in such a triangle, if each leg is \(x\), then using our simplified formula from the Pythagorean theorem, the length of the hypotenuse \(c\) becomes \( c = \sqrt{2x^2} \). This simplifies further to \( c = x\sqrt{2} \). What this means is that the hypotenuse is \(\sqrt{2}\) times any one leg's length. This consistency makes computations involving isosceles right triangles straightforward, as the ratio between the hypotenuse and a leg remains the same.

Triangle Geometry

Triangle geometry, especially involving isosceles right triangles, holds great importance in the geometric realm due to its symmetry and exact angle measures.In isosceles right triangles:

• Two sides (legs) are equal in length.
• The angles are 45°, 45°, and 90°.
• The hypotenuse is easily computed as the legs times \(\sqrt{2}\).

These triangles exhibit perfect symmetry, making them ideal for many applications, from architectural designs to simple problem-solving scenarios in mathematics. By understanding their properties, you can apply similar logic to other geometric problems, providing a strong foundation in spatial reasoning.