Problem 21 - Algebra 2 and Trigonometry

Question

The length of the shorter leg, a, of a right triangle is 6 centimeters less than the length of the hypotenuse, c, and the length of the longer leg, b, is 3 centimeters less than the length of the hypotenuse. Find the length of the sides of the right triangle.

Step-by-Step Solution

Answer

The hypotenuse is 15 cm, the shorter leg is 9 cm, and the longer leg is 12 cm.

Step 1: Define Variables

Let the hypotenuse be denoted by \(c\). The shorter leg is then \(a = c - 6\) cm, and the longer leg is \(b = c - 3\) cm.

Step 2: Apply Pythagorean Theorem

For a right triangle, the Pythagorean theorem states that \(a^2 + b^2 = c^2\). Substitute the expressions for \(a\) and \(b\):\[(c-6)^2 + (c-3)^2 = c^2\]

Step 3: Expand the Squares

Expand each of the squared terms:\[(c-6)^2 = c^2 - 12c + 36\]\[(c-3)^2 = c^2 - 6c + 9\]

Step 4: Substitute and Simplify

Substitute the expanded squares back into the equation:\[c^2 - 12c + 36 + c^2 - 6c + 9 = c^2\]Combine like terms:\[2c^2 - 18c + 45 = c^2\]

Step 5: Rearrange the Equation

Isolate the terms to form a quadratic equation:\[c^2 - 18c + 45 = 0\]

Step 6: Solve the Quadratic

Solve the equation \(c^2 - 18c + 45 = 0\) using the quadratic formula:\[c = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\]Here, \(a = 1\), \(b = -18\), \(c = 45\). Compute the discriminant and solve:Discriminant = \(324 - 180 = 144\), hence \(c = \frac{18 \pm 12}{2}\).The possible solutions are \(c = 15\) or \(c = 3\). Discard \(c = 3\) as it doesn't satisfy the problem's inequalities (it would make \(a = -3\) and \(b = 0\)).

Step 7: Find the Leg Lengths

Substitute \(c = 15\) back to find \(a\) and \(b\):\(a = 15 - 6 = 9\) cm.\(b = 15 - 3 = 12\) cm.

Key Concepts

Solving Quadratic Equations

Quadratic equations are equations of the form \(ax^2 + bx + c = 0\). Solving them typically requires finding the values of \(x\) that make the equation true. One common method to solve quadratic equations is by using the quadratic formula:

• \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\)

Here's how this works:

• Identify the coefficients (\(a\), \(b\), and \(c\)).
• Calculate the discriminant \(\Delta = b^2 - 4ac\).
• Evaluate \(x\) using the formula. The discriminant helps determine the nature of the roots: if \(\Delta > 0\), there are two real and distinct solutions; if \(\Delta = 0\), there is one real solution; and if \(\Delta < 0\), the solutions are complex numbers.

In the context of our exercise, solving the quadratic equation \(c^2 - 18c + 45 = 0\) helped find the possible lengths of the hypotenuse \(c\), ultimately leading to solving the entire triangle problem.

Right Triangle

A right triangle is a type of triangle that has one angle equal to 90 degrees. The side opposite this right angle is known as the hypotenuse, which is the longest side of the triangle. The other two sides are typically referred to as the legs.
In our exercise, we're given relationships between the triangles' sides:

• The shorter leg \(a\) is 6 centimeters less than the hypotenuse \(c\).
• The longer leg \(b\) is 3 centimeters less than the hypotenuse \(c\).

Using these relationships and applying the Pythagorean theorem, which states \(a^2 + b^2 = c^2\), allowed us to construct an equation that could be solved to find all sides of the triangle.

Expanding Algebraic Expressions

Expanding algebraic expressions involves eliminating parentheses by applying the distributive property. This is especially useful for squares of binomials like \((c-6)^2\) and \((c-3)^2\). Here's how it's done:

• The square of a binomial \((x-y)^2\) expands to \(x^2 - 2xy + y^2\).
• For \((c-6)^2\), it becomes \(c^2 - 12c + 36\).
• For \((c-3)^2\), it becomes \(c^2 - 6c + 9\).

These expanded forms were instrumental in simplifying the equation derived from the Pythagorean theorem, which involved these terms. By expanding and combining like terms, it was possible to form a straightforward quadratic equation \(c^2 - 18c + 45 = 0\) that could be solved easily.