Problem 3 - Algebra and Trigonometry with Analytic Geometry

Question

Exer. 1-8: Given the indicated parts of triangle \(A B C\) with \(\gamma=90^{\circ}\), find the exact values of the remaining parts. $$ \beta=45^{\circ}, \quad c=30 $$

Step-by-Step Solution

Answer

The remaining sides are \(a = 15\sqrt{2}\), \(b = 15\sqrt{2}\), and the remaining angle is \(\alpha = 45^{\circ}\).

Step 1: Identify Known Parts and Use the Pythagorean Theorem

We know that triangle \(ABC\) is a right triangle with \(\gamma = 90^{\circ}\). Given \(c = 30\) and \(\beta = 45^{\circ}\), we can use these to find the remaining sides and angle. Since \(\beta + \alpha = 90^{\circ}\), we have \(\alpha = 45^{\circ}\).

Step 2: Recognize Triangle Properties

This triangle is a special 45-45-90 right triangle, where both non-hypotenuse sides are equal (\(a = b\)) because their angles are both 45 degrees.

Step 3: Calculate Sides Using Triangle Properties

In a 45-45-90 triangle, the sides are in the ratio \(1:1:\sqrt{2}\). Since \(c = 30\), we use \(c = a\sqrt{2}\). Hence \(a = b = \frac{c}{\sqrt{2}} = \frac{30}{\sqrt{2}}\).

Step 4: Simplify the Sides

Simplify \(\frac{30}{\sqrt{2}}\) using rationalization. Multiply the numerator and denominator by \(\sqrt{2}\) to get \(a = b = \frac{30\sqrt{2}}{2} = 15\sqrt{2}\).

Step 5: Verify the Solution

Check the calculated sides \(a = 15\sqrt{2}\) and \(b = 15\sqrt{2}\). Use the Pythagorean Theorem to verify: \(a^2 + b^2 = c^2\). Calculate \((15\sqrt{2})^2 + (15\sqrt{2})^2 = 450\), which matches \(c^2 = 30^2 = 900\). Both calculations match, confirming correctness.

Key Concepts

45-45-90 Triangle

A 45-45-90 triangle is a special type of right triangle. It is called so because it has two angles of 45 degrees and one angle of 90 degrees.
These triangles are always isosceles, which means the two legs that meet at the right angle are the same length. This is a great feature because it simplifies calculations and helps us understand the triangle better.
The sides of a 45-45-90 triangle are in a fixed ratio. The legs, which are the two equal sides, are in the ratio of 1:1. The hypotenuse, which is the side opposite the right angle, is in the ratio of \(\sqrt{2}\) to the legs.
This ratio is vital because it helps in quickly determining the other side lengths by using one known side length, reducing the need for complex calculations.

Pythagorean Theorem

The Pythagorean Theorem is a fundamental concept in geometry. It's a tool used with right triangles to relate their side lengths. The theorem is stated as \(a^2 + b^2 = c^2\), where \(a\) and \(b\) are the legs of the right triangle and \(c\) is the hypotenuse.
In a 45-45-90 triangle, the theorem can also be employed for verification of computed values.
Once you have used the properties of a 45-45-90 triangle to determine the lengths of the legs, you can check your work using the theorem.
Simply input the values of the sides into the equation.

• If your side length calculations were correct, the Pythagorean Theorem will confirm the hypotenuse value.
• This step ensures accuracy and better understanding of how these geometrical principles intertwine.

Angle Properties

Angle properties in triangles are crucial for solving and understanding those triangles. In right triangles, the main property to remember is that the sum of the angles will always equal 180 degrees.
In a 45-45-90 triangle, we start off knowing there is a 90-degree angle. Thus, the remaining two angles are equal because the onus of being a 45-45 already tells us each of the non-right angles is 45 degrees.
The fact that the angles are add up correctly helps reassure us that we are working with the appropriate type of triangle.
Understanding angle properties is pivotal not only for solving problems but also for geometry theory in general. It supports the identification and manipulation of triangles in equations or geometric constructions.

Rationalization

Rationalization is a mathematical process used to eliminate a square root from the denominator of a fraction.
This is often necessary in trigonometry and geometry, as seen in the process of solving problems involving special triangles like the 45-45-90 triangle.
For example, if you compute the sides to be \(\frac{30}{\sqrt{2}}\), we rationalize by multiplying both numerator and denominator by \(\sqrt{2}\). This gives us \(\frac{30\sqrt{2}}{2}\), simplifying to \(15\sqrt{2}\).

• By rationalizing, calculations become easier to manage and compare, especially when checking for equivalency or performing further mathematical operations.
• It results in expressions that are clearer, cleaner, and more aligned with standard mathematical forms.

Rationalization is thus a key step in making sure that the work is presented in a conventional and acceptable format.