Problem 6 - Algebra 1: Concepts and Skills

Question

Determine whether the points are vertices of a right triangle. $$ (4,0),(4,-4),(10,-4) $$

Step-by-Step Solution

Answer

Yes, the points (4,0),(4,-4),(10,-4) form the vertices of a right triangle.

Step 1: Calculate distances between points

First, calculate the distances between the three points using the distance formula \(\sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}\), where \((x_1, y_1)\) and \((x_2, y_2)\) are the coordinates of two points. Therefore, \(AB = \sqrt{(4 - 4)^2 + (0 - (-4))^2} = 4\), \(BC = \sqrt{(4 - 10)^2 + ((-4) - (-4))^2} = 6\) and \(AC = \sqrt{(4 - 10)^2 + (0 - (-4))^2} = \sqrt{72}\)

Step 2: Apply Pythagorean theorem

Now, we apply the Pythagorean theorem to check if the square of one side length is equal to the sum of the squares of the other two sides. If we find such a combination, then we can conclude that the three points form the vertices of a right triangle. So, we check if \(AB^2 + BC^2 = AC^2\) or \(AB^2 + AC^2 = BC^2\) or \(BC^2 + AC^2 = AB^2\). Calculating will yield \(16 + 36 = 72\) or \(16 + 72 = 36\) or \(36 + 72 = 16\). Only the first equation is true \(52 = 52\), therefore the points form a right triangle.

Key Concepts

The Distance Formula

Understanding the distance formula is essential to determine the lengths between two points in coordinate geometry. It's derived from the Pythagorean theorem and essentially measures the 'straight line' distance between points on a plane.

The distance formula is given by \( d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \). In this equation, \(x_1, y_1\) and \(x_2, y_2\) represent the coordinates of two distinct points. To apply this, simply substitute the respective x and y values of your points into the formula.

For example, let's calculate the distance between points A(4,0) and B(4,-4). Plugging in the values gives us \(AB = \sqrt{(4 - 4)^2 + (0 - (-4))^2} = \sqrt{0 + 16} = \sqrt{16} = 4\) units. This formula is the cornerstone in verifying the sides of shapes like triangles on a coordinate grid.

Pythagorean Theorem in Right Triangles

The Pythagorean theorem is a fundamental principle in geometry that reveals a special relationship between the sides of a right triangle. It states that the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides.

The theorem is often written as \(a^2 + b^2 = c^2\), where \(c\) is the length of the hypotenuse, while \(a\) and \(b\) are the lengths of the other sides.

For our triangle with vertices at points (4,0), (4,-4), and (10,-4), we determined the lengths as AB=4, BC=6, and AC=\sqrt{72}. If these points form a right triangle, then applying the Pythagorean theorem should affirm that \(AB^2 + BC^2 = AC^2\), which, after calculation, indeed shows that \(16 + 36 = 52\), confirming the presence of a right angle.

Coordinates Geometry

Coordinate geometry is the study of geometric figures using the coordinate points on the Cartesian plane. Each point is determined by an ordered pair of numerical coordinates, which are the distances from the point to two fixed perpendicular directed lines, measured in the same unit of length.

When solving problems in coordinate geometry, it's essential to understand the axes, coordinates, and how to use algebra to analyze geometric shapes. For instance, in our exercise with the right triangle, knowing that the vertices can be represented as points in a coordinate system allows us to apply algebra (distance formula, Pythagorean theorem) to solve geometric problems.

Calculating Distances Between Points

Working out the distance between points on a coordinate plane is an invaluable skill. When presented with a geometric figure like a triangle, calculating these distances helps verify properties such as the triangle being right-angled.

In our exercise, after using the distance formula to find lengths AB, BC, and AC, we can check for a right angle. Given that the lengths satisfy the Pythagorean theorem's condition, we can conclude the distances between the vertices of our triangle indeed form a right angle.

In practice, accurately calculating distances leads to an understanding of the shape's properties and dimensions, making coordinate geometry a powerful tool for solving a multitude of geometric problems.