Problem 5 - Algebra 1: Concepts and Skills

Question

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

Step-by-Step Solution

Answer

No, the points (0,0), (20,0), (20,21) do not form a right triangle.

Step 1: Identify the coordinates

The coordinates given are (0,0), (20,0), (20,21).

Step 2: Calculate the distances

Use the distance formula (d = sqrt((x2-x1)^2 + (y2-y1)^2)) to measure the distances between the points. Distance between (0,0) and (20,0) is \( d_{1} = sqrt((20-0)^2 + (0-0)^2) = 20 \) units. Distance between (0,0) and (20,21) is \( d_{2} = sqrt((20-0)^2 + (21-0)^2) = sqrt(441) = 21 \) units. Distance between (20,0) and (20,21) is \( d_{3} = sqrt((20-20)^2 + (21-0)^2) = 21 \) units.

Step 3: Use Pythagorean theorem

If these points form a right triangle, they should satisfy the Pythagorean theorem. \[ (d_{1})^2 + (d_{3})^2 = (d_{2})^2 \] Substituting the values: \[ (20)^2 + (21)^2 = (21)^2 \] Simplifying gives: \[ 400 + 441 ≠ 441 \]. The result does not equal (21)^2, thus the three points do not form a right triangle.

Key Concepts

Understanding Coordinates

Coordinates are an essential part of graphing points in a two-dimensional space, typically represented with an ordered pair (x,y). These values describe a specific location on a plane where the first number represents the horizontal placement (x-coordinate) and the second number refers to the vertical placement (y-coordinate).

• You can think of coordinates like a street address. The x-coordinate indicates how far to travel horizontally, while the y-coordinate tells you how far to move vertically.
• For example, in the coordinate (0,0), both x and y are zero, placing it exactly at the origin of the coordinate plane.
• This system allows us to accurately place points in space, compare their locations, and analyze relationships between them.

Learning to interpret these ordered pairs is fundamental to working with other mathematical concepts like the distance formula and Pythagorean theorem.

Exploring the Distance Formula

The distance formula is a mathematical tool used to calculate the straight-line distance between two points in a coordinate plane. Derived from the Pythagorean theorem, this formula looks like this: \[ d = \sqrt{(x_2-x_1)^2 + (y_2-y_1)^2} \]

• Here, \((x_1, y_1)\) and \((x_2, y_2)\) represent the coordinates of the two points in question.
• The formula essentially applies the Pythagorean theorem to find the length of the hypotenuse of a right triangle formed by the horizontal and vertical distances between the points.
• To put it into practice, subtract the x-coordinates and y-coordinates separately. Square each of these values, add them together, and finally take the square root of the sum.

Using this formula can help to determine if points form specific geometric shapes, such as determining the sides of a triangle or checking if three points create a straight line.

The Pythagorean Theorem and Right Triangles

The Pythagorean theorem is a fundamental principle in mathematics given by the equation \[ a^2 + b^2 = c^2 \] This relates the lengths of the sides of a right triangle. The theorem states that in a right-angled 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.

• The sides \(a\) and \(b\) are known as the legs, and \(c\) is the hypotenuse.
• For determining whether a triangle is a right triangle, verify if the squares of the lengths of its two shorter sides sum to the square of its longest side.
• In practice, if \( a = 20 \), \( b = 21 \), and \( c = 21 \), you calculate to check if \( 20^2 + 21^2 = 21^2 \), which would indicate a right triangle. If not, as we see in the problem, there's no right triangle.

The Pythagorean theorem is indispensable in many areas of geometry and helps solve a wide range of practical and theoretical problems.