Problem 24 - Algebra and Trigonometry

Question

Solve each triangle. $$ a=3, \quad c=2, \quad B=90^{\circ} $$

Step-by-Step Solution

Answer

Right triangle does not exist with the given dimensions.

Step 1: Identify triangle type

Since angle B is given as 90°, this is a right triangle.

Step 2: Use Pythagorean theorem

For a right triangle, the Pythagorean theorem states that a² + b² = c². Given that a = 3 and c = 2, substitute the values into the theorem: a² + b² = c² becomes 3² + b² = 2².

Step 3: Simplify the equation

Simplify the equation: 9 + b² = 4.Then solve for b²:b² = 4 - 9.So, b² = -5.Since b² does not yield a real number, double-check values provided because there might be an error in the given values. A right triangle with provided dimensions does not exist in standard Euclidean geometry.

Key Concepts

Pythagorean theorem

The Pythagorean theorem is essential for solving right triangles. It is expressed as: \[ a^2 + b^2 = c^2 \]. Here, 'a' and 'b' are the legs of the triangle, and 'c' is the hypotenuse, which is the side opposite the right angle. This equation establishes a relationship between the lengths of the sides. When using the Pythagorean theorem, the sum of the squares of the two legs should equal the square of the hypotenuse. Given a right triangle with side lengths, you can use this theorem to find any missing side if the other two sides are known.

Triangle solving

When solving triangles, it is crucial to know their type. Right triangles have one angle that is exactly 90 degrees. Once you confirm it is a right triangle, you can use specific methods like the Pythagorean theorem. For the exercise given, values for a right triangle were:

• Side 'a' (leg) = 3
• Side 'c' (hypotenuse) = 2
• Angle B = 90 degrees

By substituting these values in the Pythagorean theorem, ideally, we should get: \[3^2 + b^2 = 2^2 \]. Simplifying, we reach an unexpected conclusion: \[9 + b^2 = 4 \], which then results in a non-real number for b². This indicates that there's an inconsistency in the given values.

Error checking

Error checking is vital when solving mathematical problems to avoid missteps and misunderstandings. In the exercise, after substituting the values into the theorem, we found that \[b^2 = -5 \]. This result indicated an error because the sides of a triangle in Euclidean geometry can't yield a negative square. Here are the steps to identify and correct errors in triangle problems:

• Double-check the given values to ensure they are consistent and correct.
• Verify each step and calculation diligently.
• Ensure that the results make sense in the context of Euclidean geometry.

In this problem, the given values lead to an inconsistency, suggesting there was an error in either the side lengths or the hypotenuse.

Euclidean geometry

Euclidean geometry is the study of shapes, sizes, and properties of space, particularly focusing on flat surfaces. It forms the foundation for understanding various geometric principles. Key points include:

• Right triangles form when one angle is exactly 90 degrees.
• The Pythagorean theorem applies in this geometry ensuring the relation \[a^2 + b^2 = c^2 \].

The incorrect results in the exercise highlight why consistent and accurate values are crucial in such geometric contexts. Valid right triangles should always have real, positive lengths for sides a, b, and c.