Problem 88 - Intermediate Algebra

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 (a), (b), (e), and (f) are right triangles.

Step 1: Identify the Potential Hypotenuse

In a triangle, the hypotenuse is the longest side. For each set of given numbers, identify the largest number as the potential hypotenuse \(c\).

Step 2: Check the Converse of Pythagorean Theorem

For each set, verify if \(a^2 + b^2 = c^2\) holds true. If it does, the triangle is a right triangle. Substitute \(a\), \(b\), and \(c\) with the respective side measurements and check the equation for each case.

Step 3: Verify Triangle (a)

Sides: \(9, 40, 41\). Identify \(c = 41\), \(a = 9\) and \(b = 40\). Calculate: \(9^2 + 40^2 = 81 + 1600 = 1681\) which equals \(41^2 = 1681\). This is a right triangle.

Step 4: Verify Triangle (b)

Sides: \(20, 48, 52\). Identify \(c = 52\), \(a = 20\) and \(b = 48\). Calculate: \(20^2 + 48^2 = 400 + 2304 = 2704\) which equals \(52^2 = 2704\). This is a right triangle.

Step 5: Verify Triangle (c)

Sides: \(19, 21, 26\). Identify \(c = 26\), \(a = 19\) and \(b = 21\). Calculate: \(19^2 + 21^2 = 361 + 441 = 802\) which does not equal \(26^2 = 676\). This is not a right triangle.

Step 6: Verify Triangle (d)

Sides: \(32, 37, 49\). Identify \(c = 49\), \(a = 32\) and \(b = 37\). Calculate: \(32^2 + 37^2 = 1024 + 1369 = 2393\) which does not equal \(49^2 = 2401\). This is not a right triangle.

Step 7: Verify Triangle (e)

Sides: \(65, 156, 169\). Identify \(c = 169\), \(a = 65\) and \(b = 156\). Calculate: \(65^2 + 156^2 = 4225 + 24336 = 28561\) which equals \(169^2 = 28561\). This is a right triangle.

Step 8: Verify Triangle (f)

Sides: \(21, 72, 75\). Identify \(c = 75\), \(a = 21\) and \(b = 72\). Calculate: \(21^2 + 72^2 = 441 + 5184 = 5625\) which equals \(75^2 = 5625\). This is a right triangle.

Key Concepts

Right Triangles

A right triangle is a type of triangle that contains one angle that is exactly 90 degrees (a right angle). In this triangle configuration, the side opposite the right angle is called the hypotenuse. The other two sides are simply referred to as the legs of the triangle.

The Pythagorean Theorem relates these three sides. It is expressed as: \[ a^2 + b^2 = c^2 \] where \(c\) is the hypotenuse, and \(a\) and \(b\) are the legs. This formula allows us to find the length of one side of a right triangle when the lengths of the other two are known. Remember, the Pythagorean Theorem only works for right triangles. Always make sure that you have a right triangle before applying this theorem.

Converse of Pythagorean Theorem

The converse of the Pythagorean Theorem helps us verify if a triangle is a right triangle. According to the converse, if the square of the hypotenuse equals the sum of the squares of the other two sides, then the triangle is a right triangle.

In mathematical terms, if you have three sides \(a\), \(b\), and \(c\) of a triangle, and it holds true that: \[ a^2 + b^2 = c^2 \] then you can conclude that the triangle is a right triangle. Here, \(c\) is the longest side, which logically should be the hypotenuse.

Using this converse theorem can aid in analyzing whether unconventional side lengths form a right triangle, making it a powerful tool in geometry.

Triangle Sides Analysis

When analyzing a triangle's sides to see if it is a right triangle, start by identifying the longest side as the hypotenuse candidate. This step is crucial because the Pythagorean relationship involves the hypotenuse specifically.

For each triangle:

• Identify the longest side, \(c\), as the potential hypotenuse.
• Designate the remaining two sides as \(a\) and \(b\).

Next, substitute these side lengths into the equation \(a^2 + b^2 = c^2\). If the equation holds true, the analysis confirms that these measurements correspond to a right triangle. If not, the triangle is not right angled. This methodical side analysis helps effectively verify the nature of triangles.

Geometry Verification

Geometry verification involves using mathematical principles to confirm the properties of shapes, such as verifying the type of triangle based on its side lengths.

Using the steps of verification, apply the converse of the Pythagorean Theorem:

• Select the largest side of the triangle as the hypotenuse \(c\).
• Calculate \(a^2 + b^2\) using the other two side lengths.
• Compare the calculated value with \(c^2\).

If both sides of the equation match, it confirms the triangle is right-angled. This verification process is a fundamental tactic in geometry to determine the characteristics and type of a triangle. It's not only about finding if the sides form a triangle but ensuring that it follows the predefined geometric properties.