Problem 71 - Algebra Form and Function

Question

Show that the points \((0,12),(3,0),\) and \((17 / 3,2 / 3)\) form the corners of a right triangle (that is, a triangle with a right angle).

Step-by-Step Solution

Answer

Based on the calculations, the given points (0, 12), (3, 0), and (17/3, 2/3) do not form a right triangle as the Pythagorean theorem does not hold for the three sides calculated.

Step 1: Calculate distance between Point A and Point B

We can denote the points as follows: Point A = (0, 12) Point B = (3, 0) Point C = (17/3, 2/3) The distance formula is: \(D = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}\) So, the distance between Point A and Point B is: \(D_{AB} = \sqrt{(3 - 0)^2 + (0 - 12)^2}\) \(D_{AB} = \sqrt{3^2 + (-12)^2}\) \(D_{AB} = \sqrt{9 + 144}\) \(D_{AB} = \sqrt{153}\)

Step 2: Calculate distance between Point B and Point C

Using the same distance formula, we can calculate the distance between Point B and Point C: \(D_{BC} = \sqrt{(17/3 - 3)^2 + (2/3 - 0)^2}\) \(D_{BC} = \sqrt{(8/3)^2 + (2/3)^2}\) \(D_{BC} = \sqrt{(64/9) + (4/9)}\) \(D_{BC} = \sqrt{68/9}\) \(D_{BC} = 2\sqrt{17} / 3\)

Step 3: Calculate distance between Point C and Point A

Now, let's calculate the distance between Point C and Point A: \(D_{CA} = \sqrt{(0 - 17/3)^2 + (12 - 2/3)^2}\) \(D_{CA} = \sqrt{(-17/3)^2 + (34/3)^2}\) \(D_{CA} = \sqrt{(289/9) + (1156/9)}\) \(D_{CA} = \sqrt{(1445/9)}\) \(D_{CA} = 5\sqrt{13} / 3\)

Step 4: Check if the Pythagorean theorem holds for the three sides

Now we have the lengths of the three sides, let's check if the Pythagorean theorem holds: \((D_{AB})^2 + (D_{BC})^2 = (D_{CA})^2\) \(153 + (4*17)/3 = (5\sqrt{13} / 3)^2\) \(153 + 68/3 = 25*13 / 9\) \(153*9 + 68*3 = 25*13*3\) \(1377 + 204 = 975\) \(1581 = 975\) This is not true, so the points given do not form a right triangle.

Key Concepts

Understanding the Distance Formula

The distance formula is an essential tool in geometry, helping us calculate the distance between two points on a coordinate plane. It is derived from the Pythagorean theorem and is applied by using the coordinates of the two points. This formula gives the straight-line distance, or the hypotenuse, between the two points.

Consider two points, \((x_1, y_1)\) and \((x_2, y_2)\) on a plane. To find the distance between these points, you can use the formula:

• \[D = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}\]

The formula essentially calculates the difference in the x-coordinates and y-coordinates, squares both differences, sums them, and finds the square root of the result.

Here's an application: Measure the distance between points \(A = (0, 12)\) and \(B = (3, 0)\) using the formula. Substitute the coordinates into the formula will give the distance as \[D_{AB} = \sqrt{(3 - 0)^2 + (0 - 12)^2} = \sqrt{153}\]Understanding how to use this formula is crucial, especially when deciding if lines meet at specific angles, such as determining if they form a right triangle.

Exploring the Pythagorean Theorem

The Pythagorean theorem is a fundamental concept in mathematics, often used to determine whether a triangle is a right triangle. It states that in a right triangle, the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the lengths of the other two sides.

This can be stated as:

• \(a^2 + b^2 = c^2\)

where \(c\) is the hypotenuse, and \(a\) and \(b\) are the other two sides.

In practice, after calculating all three sides of a triangle, you compare them using this relation to determine if the triangle is a right triangle. In our exercise, if \((D_{AB})^2 + (D_{BC})^2\) does not equal \((D_{CA})^2\), then it does not form a right triangle.

Using the given distances:

• \((D_{AB})^2 = 153\)
• \((D_{BC})^2 = (2\sqrt{17}/3)^2 = 68/3\)
• \((D_{CA})^2 = (5\sqrt{13}/3)^2 = 25*13/9\)

After calculating and comparing these, it was determined that these points do not satisfy the Pythagorean theorem, which confirms they do not form a right triangle.

Decoding Coordinates in Geometry

Coordinates are vital in geometry, serving as references to determine positions in a plane. Each coordinate consists of two numbers, representing the horizontal (x-axis) and vertical (y-axis) distances from a set origin point, typically represented as (0,0).

In the case of our exercise, the points given were \((0, 12), (3, 0),\) and \((17/3, 2/3)\). These points specify their locations on a Cartesian plane.

• The first point, \(A = (0, 12)\), means it's positioned "0" units to the right or left of the origin and "12" units above.
• Point \(B = (3, 0)\) is "3" units to the right of the origin and rests on the x-axis.
• Point \(C = (17/3, 2/3)\) indicates a unique position "17/3" units from the left or right and "2/3" above the x-axis.

These coordinates are crucial, allowing you to calculate distances between points and understand the dimensions of geometric shapes by applying distance formulas and theorems like the Pythagorean theorem effectively.