Problem 5 - 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. $$ a=5, \quad b=5 $$

Step-by-Step Solution

Answer

In triangle ABC, \(c = 5\sqrt{2}\) and \(\alpha = \beta = 45^{\circ}\).

Step 1: Identify Known Elements and Use Pythagorean Theorem

Triangle \(A B C\) is a right triangle with \(\gamma=90^{\circ}\). We are given sides \(a = 5\) and \(b = 5\), which are the perpendicular sides of the right triangle. To find the hypotenuse \(c\), use the Pythagorean theorem: \(c^2 = a^2 + b^2\).

Step 2: Calculate the Hypotenuse

Substitute the known values into the Pythagorean theorem: \[ c^2 = 5^2 + 5^2 = 25 + 25 = 50 \] Calculate \(c\) by taking the square root of both sides: \[ c = \sqrt{50} = \sqrt{25 \times 2} = 5\sqrt{2} \] So, the hypotenuse \(c\) is \(5\sqrt{2}\).

Step 3: Calculate Angle \(\alpha\)

Since \(\gamma = 90^{\circ}\), we use the fact that in a right triangle, the sum of angles equals \(180^{\circ}\). Therefore, \[ \alpha + \beta + \gamma = 180^{\circ} \] \[ \alpha + \beta = 90^{\circ} \]. To find \(\alpha\), note that this is an isosceles right triangle, so \(\alpha = \beta = 45^{\circ}\).

Step 4: Verify with Ratio Check for \(45^{\circ}\)

If \(\alpha = 45^{\circ}\), then \(\tan(\alpha) = \frac{a}{b} = 1\), which holds true since both sides \(a\) and \(b\) are equal.This confirms that \(\alpha = 45^{\circ}\).

Key Concepts

Right Triangle

Right triangles are special triangles where one of the angles is exactly 90 degrees. This angle is known as the right angle. In a right triangle, the side opposite the right angle is called the hypotenuse, and it is always the longest side. The other two sides, which are shorter, are referred to as the legs of the triangle. They form the right angle together.

In any geometric applications or real-life situations where right angles are involved, like in construction or navigation, right triangles offer a simple method to calculate distances or angles.
Understanding this basic framework helps in solving various problems involving trigonometry.

Pythagorean Theorem

The Pythagorean Theorem is a fundamental principle in trigonometry that applies specifically to right triangles. It states that for a right triangle, the square of the hypotenuse's length is equal to the sum of the squares of the lengths of the other two sides.

So, the theorem can be expressed with the formula: \[ c^2 = a^2 + b^2 \] where \( c \) is the hypotenuse, and \( a \) and \( b \) are the legs. This theorem is used to compute the unknown length of any side of a right triangle, provided the other two sides' lengths are known.

In our exercise, using this theorem allowed us to find the hypotenuse as \( 5\sqrt{2} \). This highlights its utility in calculations involving right triangles.

Isosceles Right Triangle

An isosceles right triangle is a unique type of right triangle. It has two equal sides, which also makes two of its angles equal. In this triangle, the angles opposite the equal sides are each 45 degrees.

The fact that the sides \( a \) and \( b \) are of equal lengths indicates that triangle \( ABC \) is an isosceles right triangle. Consequently, this also means that the angles \( \alpha \) and \( \beta \) are both \( 45^{\circ} \).

Understanding the properties of an isosceles right triangle can simplify geometric calculations and ensure that angle measures are correctly identified, aiding in quick problem-solving.

Angles in Triangle

Understanding the angles in a triangle is crucial for solving geometric problems. The sum of angles in any triangle is always 180 degrees. In the case of a right triangle, one of these angles is a fixed \( 90^{\circ} \).

Knowing that, you can always find the remaining angles if at least one other angle is known. Specifically, for the isosceles right triangle, the two non-right angles will each measure \( 45^{\circ} \).

This rule helps in verifying the characteristics of triangles and deducing unknown angles using the known ones. It underscores the coherent geometric principle where all shapes and angles align cohesively to sum up to a systematic total.