Problem 33 - Algebra 1

Question

Graph the points. Decide whether they are vertices of a right triangle. $$(0,-4),(4,-1),(4,-4)$$

Step-by-Step Solution

Answer

The points (0,-4), (4,-1), and (4,-4) when plotted, actually form a right triangle. The triangle is right-angled at the point (4,-4). This was found by verifying Pythagoras theorem as well as checking the slope of the lines.

Step 1: Plot the Points

Start by plotting the points (0,-4), (4,-1), and (4,-4) on a Cartesian plane. Draw lines to connect these points to form a triangle.

Step 2: Calculate Slopes

Calculate the slope of the line segment connecting the points (0,-4) and (4,-1), then calculate the slope of the line segment connecting (4,-1) and (4,-4). If these slopes are negative reciprocals of each other, then they form a right angle.

Step 3: Evaluate

If the two lines are not perpendicular (their slopes are not negative reciprocals), calculate the lengths of all three sides of the triangle using the distance formula and apply Pythagoras theorem to verify if it is a right triangle. If the sum of the squares of the lengths of two of the sides equal the square of the length of the third side, we have a right triangle.

Key Concepts

Cartesian Plane

The Cartesian Plane is a two-dimensional plane defined by two perpendicular axes: the x-axis (horizontal) and the y-axis (vertical). It's like a map where we plot points using pairs of numbers called coordinates. Each coordinate pair \( x, y \) tells us exactly where a point is located on this grid. For example, the point (0, -4) is situated at 0 on the x-axis and -4 on the y-axis. Every move to the right or upwards involves using positive numbers, while moves to the left or down are represented by negative ones.
The Cartesian Plane helps us visualize problems and solutions in geometry, including plotting points and shapes like triangles. When we plot the points (0,-4), (4,-1), and (4,-4), we can visually assess the shape they form, aiding us in tasks like determining whether a triangle is a right triangle. It also becomes our playground for further mathematical calculations needed to answer complex questions about shapes and lines.

Slope Calculation

When dealing with lines in geometry, slope is a crucial concept. The slope of a line tells us how steep the line is, and it is calculated by determining the "rise over run." This means the change in the y-values divided by the change in the x-values between two points on the line.
To find the slope of the line segment between the points (0,-4) and (4,-1), we use the formula: \( m = (y_2-y_1) / (x_2-x_1) \)
Filling in the coordinates gives us: \( m = (-1-(-4))/(4-0) = 3/4 \). For the line segment between (4,-1) and (4,-4), we calculate: \( m = (-4-(-1))/(4-4) \). Since the change in the x-values is zero, this line is vertical, and a vertical line has an undefined slope.
Understanding slope helps in determining if two lines are perpendicular. Lines are perpendicular if their slopes are negative reciprocals. However, one of our lines is vertical, which automatically means it is perpendicular to any horizontal line, hinting at a potential right triangle.

Distance Formula

The Distance Formula is used to calculate the distance between two points in the Cartesian Plane. Imagine you want to measure the length of each side of a triangle; this formula lets you do that quickly. The formula is an application of the Pythagorean Theorem, adapted for points on a plane: \( d = \sqrt{(x_2-x_1)^2 + (y_2-y_1)^2} \).
For example, to find the length of the side between points (0,-4) and (4,-1), plug in their coordinates: \( d = \sqrt{(4-0)^2 + (-1-(-4))^2} = \sqrt{4^2 + 3^2} = \sqrt{16 + 9} = \sqrt{25} = 5 \).
This calculation shows how the Distance Formula helps establish side lengths, crucial for confirming the characteristics of a right triangle. Calculating these side lengths is essential when using the Pythagorean Theorem to determine right-angle relationships in triangles.

Pythagorean Theorem

The Pythagorean Theorem is a fundamental principle in geometry, particularly when analyzing right triangles. It states that in a right 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: \( c^2 = a^2 + b^2 \).
To determine if our vertices from the Cartesian plane form a right triangle, after computing side lengths using the Distance Formula, we check if this condition holds true. If two sides squared and summed equal the hypotenuse squared, then it confirms the presence of a right angle.
This theorem not only helps in validating the structure of a triangle but also in solving various real-world problems where right angles and precise measurements are involved. It's a powerful tool in both theoretical and practical applications of mathematics.