Problem 8 - Algebra 1

Question

Decide whether the points are vertices of a right triangle. \((-2,0),(-1,0),(1,7)\)

Step-by-Step Solution

Answer

No, the given points are not vertices of a right triangle.

Step 1: Calculation of distances

Calculate the distances between each pair of points. Let's denote the points as A(-2,0), B(-1,0), and C(1,7). Using the distance formula \(d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}\), we get distances AB = \( \sqrt{ (-1 - (-2) )^2 + 0 - 0 }) = 1 \), AC = \( \sqrt{ (1 - (-2) )^2 + (7 - 0) )^2 } = \sqrt{9 + 49} = \sqrt{58} \), and BC = \( \sqrt{ (1 - (-1) )^2 + (7 - 0) )^2 } = \sqrt{ 4 + 49 } = \sqrt{53} \)

Step 2: Using the Pythagorean theorem

According to the Pythagorean theorem, for a right triangle the square of the hypotenuse (the longest side) should be equal to the sum of squares of the other two sides. Check this by squaring each of the calculated lengths, then confirm if any two sides' squares sum to the square of the third side.

Step 3: Verification

Let's square each of the calculated lengths: \(AB^2 = 1^2 = 1\), \(AC^2 = (\sqrt{58})^2 = 58\), \(BC^2 = (\sqrt{53})^2 = 53\). None of the lengths squared sums up to the square of the remaining length, so these points do not form a right triangle.

Key Concepts

Distance Formula

The distance formula is a key mathematical concept used to calculate the straight line distance between two points in coordinate geometry. Imagine plotting two points on a graph with X and Y coordinates assigned to each. The formula is derived from the Pythagorean theorem which links the sides of a right triangle to its hypotenuse. The distance between points A \big((x_1, y_1)\big) and B \big((x_2, y_2)\big) is given by:
d = \(\sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}\)
This equation essentially tells us the length of the 'line segment' connecting A and B. When solving problems involving the distance between coordinates, pay careful attention to subtracting and squaring the correct values to avoid negative distances, which are not possible in the context of measuring lengths.

Pythagorean Theorem

At the heart of understanding right triangles is the Pythagorean theorem, a fundamental principle in geometry. It states that in a right triangle, the square of the length of the hypotenuse is equal to the sum of the squares of the lengths of the other two shorter sides. Represented algebraically, if 'c' is the hypotenuse, and 'a' and 'b' are the other two sides, the theorem can be expressed as:\(c^2 = a^2 + b^2\)
This theorem is not just theoretical; it's widely used in various real-world applications from construction to navigation. When applying this theorem, identify the longest side correctly to test for a right triangle. Misinterpreting the sides can lead to incorrect conclusions, so always ensure the hypotenuse is in fact the longest side before proceeding with the Pythagorean theorem.

Coordinate Geometry

Where lines and shapes meet numbers and calculations—that's where coordinate geometry, also known as analytic geometry, comes into play. It involves the study of geometric figures graphically represented using the Cartesian coordinate system. Every point is defined by a pair of numerical coordinates, which are distances to two or three mutually perpendicular lines, called axes.

Understanding the Axes

Coordinate geometry is built on two main axes: the horizontal x-axis and the vertical y-axis. They intersect at a point known as the origin. By combining our knowledge of the distance formula and the Pythagorean theorem, we can locate points, calculate distances, and determine the nature of shapes in this two-dimensional space.
In practical exercises, such as determining whether a set of points forms a right triangle, coordinate geometry aids in calculating distances and applying the Pythagorean theorem to analyze triangle characteristics. The synergy between algebra and geometry in coordinate geometry opens the door to solving many practical problems with precision and elegance.