Question

Decide whether the points are vertices of a right triangle. \((0,0),(20,0),(20,21)\)

Short answer

The points (0,0),(20,0),(20,21) do not form a right triangle.

Step-by-step solution

Identify the distances between the points

First calculate the distance of the line segments forming among the given points. We use the distance formula considering two points \((x1,y1)\) and \((x2,y2)\) given by: \(\sqrt{(x2-x1)^2 + (y2-y1)^2}\). The distance between (0,0) and (20,0) is 20 units, between (20,0) and (20,21) is 21 units, and between (0,0) and (20,21) is \(\sqrt{((20-0)^2 + (21-0)^2)}= \sqrt{441} = 21\), which is the hypotenuse.

Use the Pythagorean theorem

Use the Pythagorean Theorem to check if \(a^2 + b^2 = c^2\). Take the distances calculated in Step 1 as a, b, and c. So substitute the distances into the Pythagorean theorem, we get \(20^2 + 21^2 = 21^2\) after calculation we get, \(400 + 441 = 441\). This equation does not hold true. Therefore, the points do not form a right triangle.

Key concepts

Right Triangle

Understanding the concept of a right triangle is crucial when solving problems related to geometry and trigonometry. A right triangle is a special type of triangle where one angle is exactly 90 degrees, known as the right angle. The sides forming this right angle are called the 'legs', and the side opposite the right angle is the 'hypotenuse', the longest side of the triangle. In the context of the exercise, we examine whether the given points form a right triangle by checking the lengths of the triangle's sides.

It's important to note that for a set of points to represent the vertices of a right triangle, the distances between the points must satisfy the Pythagorean theorem. This theorem, which you will read more about below, allows us to test for the right angle by using the lengths of the sides.

Distance Formula

The distance formula is an essential tool in coordinate geometry that enables us to calculate the distance between two points in the Cartesian plane. The premise of the formula derives from the Pythagorean theorem and is expressed as: \(d = \sqrt{(x_2-x_1)^2 + (y_2-y_1)^2}\), where \(d\) represents the distance between two points with coordinates \(x_1, y_1\) and \(x_2, y_2\).

Application in the Exercise

Applying the distance formula to the exercise, we calculated the lengths of the sides formed by the given points. This computation is the first step in determining whether these points can be vertices of a right triangle. If you're faced with a geometry problem requiring distance calculations, remember the distance formula is your go-to equation to find the length of a line segment between any two points on a graph using their coordinates.

Coordinates

Coordinates are a set of values that show an exact position on a two-dimensional plane using an ordered pair. They are essential in identifying locations in geometry and many applied fields like GIS and computer graphics. The first number, known as the x-coordinate, corresponds to a point's position along the horizontal axis, while the second number, the y-coordinate, relates to the vertical axis.

In the context of the given problem, we deal with three sets of coordinates representing the vertices of a triangle. By using these coordinates, we apply the distance formula mentioned above to establish the lengths of the sides of the triangle. It's the combination of a precise understanding of coordinates and the ability to apply the distance formula that enables us to tackle such geometric problems effectively.