Problem 23 - Prealgebra

Question

The lengths of three sides of a triangle are given. Determine whether each triangle is a right triangle. $$a=24, b=28, c=32$$

Step-by-Step Solution

Answer

The triangle with sides 24, 28, and 32 is not a right triangle.

Step 1: Understand the Problem

We need to determine if a triangle with sides of lengths 24, 28, and 32 is a right triangle. A right triangle satisfies the Pythagorean theorem, which states that the square of the length of the hypotenuse (longest side) is equal to the sum of the squares of the other two sides.

Step 2: Identify the Longest Side

In any triangle, the longest side is considered the potential hypotenuse when checking for a right triangle. Here, the longest side is 32.

Step 3: Apply the Pythagorean Theorem

Check if the Pythagorean theorem holds for the triangle by calculating: \( c^2 = a^2 + b^2 \). Substitute \( a = 24 \), \( b = 28 \), and \( c = 32 \):\[ 32^2 = 24^2 + 28^2 \]\[ 1024 = 576 + 784 \]\[ 1024 = 1360 \]

Step 4: Conclusion

Since \( 1024 eq 1360 \), the equation does not hold, meaning the given triangle is not a right triangle.

Key Concepts

Right Triangle

A right triangle is a type of triangle that has a special property: one of its angles is exactly 90 degrees. This is a characteristic that makes right triangles unique. When the shortest sides of a triangle ('legs') and the longest side ('hypotenuse') fit the Pythagorean theorem, you have a right triangle.
Pythagorean theorem is the key to identifying whether a triangle is a right triangle, either by measuring angles or by calculating side lengths. In mathematical terms, the theorem states:

• The square of the hypotenuse (longest side) is the sum of the squares of the other two sides.
• This is expressed with the formula: \[ c^2 = a^2 + b^2 \]

For the triangle with sides 24, 28, and 32, we hypothesize that 32 is the hypotenuse. Plugging into the theorem shows the equation does not balance, indicating no right angle lurks in these measurements.

Triangle Properties

Triangles are foundational shapes in geometry, defined as figures with three sides and three angles. When discussing triangle properties, we focus on their angles and sides to understand their shape and classification.
Triangles come in several types, such as equilateral, isosceles, or scalene, which are defined based on the equivalency or differences in their sides and angles. A key property is that the sum of a triangle's internal angles is always 180 degrees.
In solving problems like determining a right triangle, we look at the side lengths and apply mathematical principles like the Pythagorean theorem for validation. Each triangle type follows its distinct properties, but the tool to determine if a triangle is a right one is consistent, checking the relationship between its sides.

Geometry

Geometry is the mathematical study of shapes, sizes, and the properties of space. It's a branch where understanding the relationships between different shapes, especially triangles, is key.
It is through geometric principles and equations that we solve real-life problems involving space and form. Geometry deals with different shapes' areas, perimeters, and angles. Triangles, in particular, are vital because they appear in various configurations—both practical and theoretical.
Right triangles specifically play a crucial role in geometry due to their defined sides and angles. They often simplify complex shapes and calculation processes. Problems like determining a triangle's type show how geometric rules help understand the underlying form. Through geometry, we visualize and derive solutions much like how we determine if 24, 28, and 32 can truly form a right triangle.