Problem 53 - Algebra A Combined Function

Question

Solve. The hypotenuse of an isosceles right triangle is 2 centimeters longer than either of its legs. Find the exact length of each side. (Hint: An isosceles right triangle is a right triangle whose legs are the same length.)

Step-by-Step Solution

Answer

Each leg is \(2 + 2\sqrt{2}\) cm, and the hypotenuse is \(4 + 2\sqrt{2}\) cm.

Step 1: Define Variables

Let \( x \) represent the length of each leg of the isosceles right triangle. Because the hypotenuse is 2 centimeters longer than each leg, we can express this as \( x + 2 \).

Step 2: Use the Pythagorean Theorem

The Pythagorean Theorem for a right triangle is given by \( a^2 + b^2 = c^2 \), where \( a \) and \( b \) are the legs and \( c \) is the hypotenuse. Here, both legs \( a \) and \( b \) are \( x \), so we have \( x^2 + x^2 = (x + 2)^2 \).

Step 3: Simplify and Set Up Equation

Simplify \( x^2 + x^2 = (x + 2)^2 \) to get \( 2x^2 = (x + 2)^2 \). Expanding \( (x + 2)^2 \), we have:\[(x + 2)^2 = x^2 + 4x + 4\]So, the equation becomes:\[2x^2 = x^2 + 4x + 4\]

Step 4: Solve the Quadratic Equation

Subtract \( x^2 \) from both sides to get:\[x^2 = 4x + 4\]Move all terms to one side to form a quadratic equation:\[x^2 - 4x - 4 = 0\]

Step 5: Apply the Quadratic Formula

The quadratic formula is given by \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \) for an equation \( ax^2 + bx + c = 0 \). Here, \( a = 1 \), \( b = -4 \), and \( c = -4 \).Calculate the discriminant:\[b^2 - 4ac = (-4)^2 - 4(1)(-4) = 16 + 16 = 32\]Substitute into the quadratic formula:\[x = \frac{-(-4) \pm \sqrt{32}}{2(1)} = \frac{4 \pm \sqrt{32}}{2} = \frac{4 \pm 4\sqrt{2}}{2}\]This simplifies to:\[x = 2 \pm 2\sqrt{2}\]

Step 6: Determine Valid Solution

Since the length cannot be negative, we choose the positive solution from \( x = 2 \pm 2\sqrt{2} \), which is \( x = 2 + 2\sqrt{2} \).

Step 7: Length of Hypotenuse

The length of the hypotenuse is \( x + 2 \), so substitute the value of \( x \):Hypotenuse = \((2 + 2\sqrt{2}) + 2 = 4 + 2\sqrt{2}\)

Key Concepts

Isosceles Right Triangle

An isosceles right triangle is a fascinating shape in geometry. It has a unique feature where two sides, called the legs, are of equal length. This special property makes it isosceles. Moreover, it is a right triangle, meaning it has one angle that is exactly 90 degrees.
This kind of triangle follows the Pythagorean Theorem where the hypotenuse (the longest side opposite the right angle) relates to the legs via the equation:

• If each leg has a length of \( x \), the hypotenuse will be \( x\sqrt{2} \).
• In this problem, as a hint, the hypotenuse is 2 centimeters longer than the legs, so there’s a little twist!

Recognizing these triangles helps us calculate other dimensions easily when at least one length is known.

Quadratic Equation

Quadratic equations pop up in many mathematical problems, especially when dealing with shapes. They are expressed in the form \( ax^2 + bx + c = 0 \).

• In the context of our isosceles right triangle, finding the length of the sides turns into a quadratic equation.
• When we use the Pythagorean Theorem here, we get \( 2x^2 = (x + 2)^2 \). By simplifying and rearranging, this becomes a quadratic equation \( x^2 - 4x - 4 = 0 \).
• This step is crucial because it sets the stage for finding the unknown \( x \), the length of each leg.

Understanding and solving quadratic equations are essential skills in tackling geometric problems like this.

Quadratic Formula

The quadratic formula is a fundamental tool to solve equations that appear in the general quadratic form \( ax^2 + bx + c = 0 \). It is given by:

• \[x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\]
• This formula provides solutions to any quadratic equation, taking the coefficients \( a \), \( b \), and \( c \) into account.

For our triangle:

• We have \( a = 1 \), \( b = -4 \), and \( c = -4 \). The discriminant, \( b^2 - 4ac \), becomes 32, indicating two real solutions.
• Using the quadratic formula, the solutions are \( x = 2 \pm 2\sqrt{2} \).
• Since a side cannot be negative, we choose \( x = 2 + 2 \sqrt{2} \) for the leg's length.

This demonstrates how the quadratic formula aids in solving practical problems.

Length of Sides

Determining the length of each side in a geometric problem is often the ultimate goal. In our isosceles right triangle, once the necessary calculations are done:

• The legs, being equal, each have a length of \( x = 2 + 2\sqrt{2} \).
• The hypotenuse, which is 2 centimeters longer than the legs, measures \( 4 + 2 \sqrt{2} \).

Knowing how to find these lengths involves using both the properties of the triangle and the results from solving the quadratic equation.

• These calculations allow us to get precise measurements, crucial in geometric design and application.
• Understanding these results also gives deeper insight into the symmetrical and predictable nature of isosceles right triangles.

Getting these lengths correct builds a solid foundation for more complex mathematical challenges.