Problem 96 - Algebra for College Students

Question

The converse of the Pythagorean theorem is also true. It states, "If the measures \(a, b\), and \(c\) of the sides of a triangle are such that \(a^{2}+b^{2}=c^{2}\), then the triangle is a right triangle with \(a\) and \(b\) the measures of the legs and \(c\) the measure of the hypotenuse." Use the converse of the Pythagorean theorem to determine which of the triangles with sides of the following measures are right triangles. (a) \(9,40,41\) (b) \(20,48,52\) (c) \(19,21,26\) (d) \(32,37,49\) (e) \(65,156,169\) (f) \(21,72,75\)

Step-by-Step Solution

Answer

Triangles 9,40,41; 20,48,52; 65,156,169; and 21,72,75 are right triangles.

Step 1: Understanding the Problem

We need to determine if the given triangles with sides \(a, b,\) and \(c\) satisfy the converse of the Pythagorean theorem. For a triangle to be a right triangle, the equation \(a^2 + b^2 = c^2\) must be satisfied, where \(c\) is the largest side (hypotenuse), and \(a\) and \(b\) are the other two sides.

Step 2: Evaluate Triangle (a)

For sides \(9, 40, 41\), first identify the largest side as the hypotenuse \(c = 41\). Check if \(9^2 + 40^2 = 41^2\). Calculate: \(9^2 = 81\), \(40^2 = 1600\), \(41^2 = 1681\). So, \(81 + 1600 = 1681\), which confirms \(a^2 + b^2 = c^2\). Thus, it is a right triangle.

Step 3: Evaluate Triangle (b)

For sides \(20, 48, 52\), the largest side is \(c = 52\). Check if \(20^2 + 48^2 = 52^2\). Calculate: \(20^2 = 400\), \(48^2 = 2304\), \(52^2 = 2704\). So, \(400 + 2304 = 2704\), confirming \(a^2 + b^2 = c^2\). Thus, it is a right triangle.

Step 4: Evaluate Triangle (c)

For sides \(19, 21, 26\), the largest side is \(c = 26\). Check if \(19^2 + 21^2 = 26^2\). Calculate: \(19^2 = 361\), \(21^2 = 441\), \(26^2 = 676\). So, \(361 + 441 = 802\), which is not equal to \(676\). It is not a right triangle.

Step 5: Evaluate Triangle (d)

For sides \(32, 37, 49\), the largest side is \(c = 49\). Check if \(32^2 + 37^2 = 49^2\). Calculate: \(32^2 = 1024\), \(37^2 = 1369\), \(49^2 = 2401\). So, \(1024 + 1369 = 2393\), which is not equal to \(2401\). It is not a right triangle.

Step 6: Evaluate Triangle (e)

For sides \(65, 156, 169\), the largest side is \(c = 169\). Check if \(65^2 + 156^2 = 169^2\). Calculate: \(65^2 = 4225\), \(156^2 = 24336\), \(169^2 = 28561\). So, \(4225 + 24336 = 28561\), confirming \(a^2 + b^2 = c^2\). Thus, it is a right triangle.

Step 7: Evaluate Triangle (f)

For sides \(21, 72, 75\), the largest side is \(c = 75\). Check if \(21^2 + 72^2 = 75^2\). Calculate: \(21^2 = 441\), \(72^2 = 5184\), \(75^2 = 5625\). So, \(441 + 5184 = 5625\), confirming \(a^2 + b^2 = c^2\). Thus, it is a right triangle.

Key Concepts

Converse of the Pythagorean Theorem

The Pythagorean Theorem is a fundamental concept in geometry that deals with right triangles. The theorem states that in a right triangle, the sum of the squares of the two shorter sides (\(a\) and \(b\)) is equal to the square of the longest side (the hypotenuse, \(c\)): \[a^2 + b^2 = c^2.\]
The converse of the Pythagorean Theorem states that if three sides of a triangle satisfy this equation, then the triangle must be a right triangle. This means, given any triangle, you can determine if it is a right triangle by verifying this condition.

• Identify the longest side of the triangle as the hypotenuse \(c\).
• Verify whether the equation \(a^2 + b^2 = c^2\) holds true using algebraic calculations.

If the equation holds, you can confidently say the triangle is right; otherwise, it's not. This is a powerful tool for confirming triangle properties in geometry.

Right Triangles

Right triangles are a special category of triangles characterized by having one 90-degree angle, known as the right angle. The defining feature of a right triangle is the relationship between its sides, given by the Pythagorean Theorem.
In a right triangle:

• The longest side, opposite the right angle, is called the hypotenuse.
• The other two sides are known as the legs.

A right triangle has unique properties that make it a cornerstone in geometry:

• It can be used to derive and understand trigonometric ratios, such as sine, cosine, and tangent.
• It allows calculations of distances and heights using simple algebraic formulas.

Recognizing a right triangle helps simplify problems, making it easier to apply further mathematical concepts.

Triangle Properties

Triangles are three-sided polygons with a few important properties distinct to their types, including scalene, isosceles, and equilateral triangles. For any type of triangle, three main principles apply:

• The sum of the internal angles always equals 180 degrees.
• The lengths of any two sides must be greater than the length of the third side (triangle inequality theorem).

Right triangles obey these universal laws but with the additional characteristic defined by the Pythagorean Theorem. For identifying a right triangle, not only do we need to check these basic properties, but also verify the squared sum equality of its two legs amounting to the square of its hypotenuse.
Understanding these properties is crucial for solving geometric problems, as they define the possible dimensions and shape of the triangle based on side lengths and angles.

Algebra

Algebra is key in solving geometrical problems involving triangles, particularly with the Pythagorean Theorem. It provides a systematic approach to verify whether a set of sides can form a right triangle. The algebraic manipulation involves:

• Identifying which side is the lengthiest so it can be considered the hypotenuse, \(c\).
• Computing the squares of each side and checking if \(a^2 + b^2 = c^2\).

By systematically carrying out these calculations, algebra ensures accuracy in determining which sets of sides conform to the properties of right triangles. This method also demonstrates the practical application of algebraic principles to solve real-world problems related to shapes and measurements. Understanding this relationship enhances problem-solving skills, critical for succeeding in both mathematics and applied sciences.