Problem 13 - Intermediate Algebra

Question

Verify that the points \((-3,1),(5,7)\), and \((8,3)\) are vertices of a right triangle. [Hint: If \(a^{2}+b^{2}=c^{2}\), then it is a right triangle with the right angle opposite side \(c\).]

Step-by-Step Solution

Answer

Yes, the points form a right triangle with the right angle opposite the side with length 10.

Step 1: Determine Distance Between Points (−3, 1) and (5, 7)

Use the distance formula: \(d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}\). Here, \((x_1, y_1) = (-3, 1)\) and \((x_2, y_2) = (5, 7)\). Calculate as follows:\[d = \sqrt{(5 - (-3))^2 + (7 - 1)^2} = \sqrt{8^2 + 6^2} = \sqrt{64 + 36} = \sqrt{100} = 10.\]

Step 2: Determine Distance Between Points (5, 7) and (8, 3)

Use the distance formula again with \((x_1, y_1) = (5, 7)\) and \((x_2, y_2) = (8, 3)\):\[d = \sqrt{(8 - 5)^2 + (3 - 7)^2} = \sqrt{3^2 + (-4)^2} = \sqrt{9 + 16} = \sqrt{25} = 5.\]

Step 3: Determine Distance Between Points (8, 3) and (−3, 1)

Again, apply the distance formula with \((x_1, y_1) = (8, 3)\) and \((x_2, y_2) = (-3, 1)\):\[d = \sqrt{(-3 - 8)^2 + (1 - 3)^2} = \sqrt{(-11)^2 + (-2)^2} = \sqrt{121 + 4} = \sqrt{125}.\]

Step 4: Verify Pythagorean Theorem for Right Triangle

Identify the longest distance as \(c\) and check if it satisfies \(a^2 + b^2 = c^2\) with the other two distances:- Shortest distances are 5 (between points (5, 7) and (8, 3)) and 10 (between points (-3, 1) and (5, 7)).- Calculate: \(5^2 + 10^2 = 25 + 100 = 125\).- Check against longest distance squared: \((\sqrt{125})^2 = 125\).The points do satisfy the Pythagorean Theorem.

Key Concepts

Distance Formula

The distance formula is a mathematical tool used to determine the distance between two points in a plane. Think of it like a ruler for points on a grid; it helps us find out how far apart they are. The formula is derived from the Pythagorean theorem and is expressed as follows:

• For two given points, \((x_1, y_1)\) and \((x_2, y_2)\), the distance \(d\) is calculated using \(d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}\).

Why is this helpful? It provides a straightforward way to calculate distances when you have the coordinates of the points. For instance, if you know two houses on a map are at coordinates \((3, 4)\) and \((7, 1)\), the distance formula lets you know exactly how far apart they are.
By substituting the coordinates into the formula, we compute the distance directly.
This method is especially useful in coordinate geometry, where calculating the distances helps verify properties of geometric shapes like triangles or rectangles.

Pythagorean Theorem

The Pythagorean Theorem is a fundamental principle in geometry, particularly related to right triangles. It states that in a right triangle, the square of the length of the hypotenuse (the longest side) is equal to the sum of the squares of the lengths of the other two sides. Mathematically expressed as:

• \(c^2 = a^2 + b^2\)

where \(c\) is the hypotenuse, and \(a\) and \(b\) are the other two sides.
This theorem not only applies to geometrical shapes but is also extremely useful in various real-world applications. For example, it can be used in building construction where you need to ensure a structure has precisely right angles.
In the context of coordinate geometry, the theorem aids in verifying if a set of three points forms a right triangle by ensuring the longest side squared equals the sum of the squares of the shorter sides.
This is what we call the process of right triangle verification.

Coordinate Geometry

Coordinate geometry, also known as analytic geometry, is the study of geometry using a coordinate system. This branch of mathematics bridges the gap between algebra and geometry through graphs and coordinates.
With coordinate geometry, you can explore geometric problems and verify geometric properties using algebraic methods.
In essence, it involves describing geometric figures like points, lines, and shapes using algebraic equations.

• Each point is defined by an \((x, y)\) coordinate pair, which makes it easy to calculate distances and angles.
• Understanding the position and relation of shapes becomes simpler, thanks to this systematic approach.

This method allows you to transform abstract geometric concepts into practical, computable tasks by using coordinates and formulas.
For students, coordinate geometry is particularly beneficial because it visualizes algebraic equations, helping to better understand and solve problems.
It turns geometric problems into sets of algebraic equations, making complex calculations manageable and verifiable, especially useful in proofs like verifying a right triangle.