Problem 3 - Precalculus: Functions and Graphs

Question

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

\( a = b = 15\sqrt{2}, \gamma = 90^{\circ}, \alpha = 45^{\circ} \).

Step 1: Identify Triangle Type

In triangle ABC, given that \( \gamma = 90^{\circ} \), we know that this triangle is a right-angled triangle.

Step 2: Apply Trigonometric Functions

Use the property of right-angled triangles that states the sum of all angles is \( 180^{\circ} \). Thus, \( \alpha + \beta + \gamma = 180^{\circ} \). Since \( \gamma = 90^{\circ} \), \( \beta = 45^{\circ} \), we find \( \alpha = 180^{\circ} - 90^{\circ} - 45^{\circ} = 45^{\circ} \).

Step 3: Determine Triangle Configuration

Since \( \alpha = 45^{\circ} \) and \( \beta = 45^{\circ} \), triangle ABC is an isosceles right-angled triangle, implying that the two legs (\( a \) and \( b \)) are equal.

Step 4: Use Pythagorean Theorem

For right-angled triangles, \( c^2 = a^2 + b^2 \). Given \( b = a \), \( c^2 = 2a^2 \).

Step 5: Solve for a and b

Given \( c = 30 \), substitute into the equation: \( 30^2 = 2a^2 \). Therefore, \( 900 = 2a^2 \), which implies \( a^2 = 450 \) and \( a = \sqrt{450} = 15\sqrt{2} \). Since \( a = b \), \( b = 15\sqrt{2} \).

Key Concepts

Isosceles Triangle

An isosceles triangle is one where two of its three sides are equal in length. This special property leads to two angles also being equal. In our right-angled triangle ABC, where we have \( \alpha = 45^{\circ} \) and \( \beta = 45^{\circ} \), the triangle is not only right-angled because of the \( 90^{\circ} \) angle at \( \gamma \), but also isosceles. This ensures that the two sides opposite the \( 45^{\circ} \) angles are equal. Here are key points to remember about isosceles triangles:

• Two sides are equal: This means in triangle ABC, \( a = b \).
• Two angles are equal: In our triangle, both \( \alpha \) and \( \beta \) are \( 45^{\circ} \).
• Common in right-angled triangles, especially those with \( 45^{\circ} \) angles.

The basic symmetry of isosceles triangles makes calculations easier, as seen in this exercise where knowing two angles allows us to determine the third angle and the equal side lengths.

Pythagorean Theorem

The Pythagorean Theorem applies to right-angled triangles and is essential for finding side lengths. It states that the square of the length of the hypotenuse \( c \) is equal to the sum of the squares of the other two sides \( a \) and \( b \). In formula terms, it is written as:\[c^2 = a^2 + b^2\]In our case with triangle ABC:

• The hypotenuse \( c = 30 \).
• Given it’s an isosceles right triangle, \( a = b \).
• Using the equation, substitute \( b = a \), leading to \( c^2 = 2a^2 \).
• Solving this, \( 30^2 = 2a^2 \), simplifies to \( a^2 = 450 \), resulting in \( a = 15\sqrt{2} \).

The Pythagorean Theorem is powerful because it provides a straightforward way to compute unknown side lengths when at least one side length is known. Through it, we easily find that \( a \) and \( b = 15\sqrt{2} \) when \( c = 30 \).

Angle Sum Property

The angle sum property is fundamental in triangles. It states that the sum of the angles in any triangle is always \( 180^{\circ} \). When solving problems involving triangles, this property is indispensable. For triangle ABC:

• We have \( \gamma = 90^{\circ} \), making it a right-angle triangle.
• Given \( \beta = 45^{\circ} \), we use the angle sum property to find \( \alpha \).
• The calculation is \( \alpha = 180^{\circ} - 90^{\circ} - 45^{\circ} = 45^{\circ} \).

With the angle sum property, even if only two angles are known, finding the third becomes a simple subtraction task. Hence, it confirms that triangle ABC is not just right-angled, but also perfectly isosceles due to its equal angles, reflecting its equal side lengths.