Problem 7 - Algebra 1

Question

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

Step-by-Step Solution

Answer

No, the points \((4,0),(4,-4),(10,-4)\) do not form a right triangle.

Step 1: Calculate Distances

Calculate distances between points to find the lengths of the triangle sides. Distances are calculated using distance formula \(d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}\) \n Distance from \((4,0)\) to \((4,-4)\) is \(d_1 = \sqrt{(4 - 4)^2 + (0 - -4)^2} = 4\) \n Distance from \((4,0)\) to \((10,-4)\) is \(d_2 = \sqrt{(10 - 4)^2 + (0 - -4)^2} = \sqrt{36+16} = 6\sqrt{2}\) \n Distance from \((4,-4)\) to \((10,-4)\) is \(d_3 = \sqrt{(10 - 4)^2 + (-4 - -4)^2} = 6\)

Step 2: Check Pythagorean theorem

To check if these points could form a right triangle, try to validate the Pythagorean theorem, which is \(a^2 + b^2 = c^2\), where \(c\) is the longest side. For this, identify the longest side which appears to be \(c = d_2 = 6\sqrt{2}\) and the other two sides are \(a = d_1 = 4\) and \(b = d_3 = 6\). Thus, \((4)^2 + (6)^2 = 16 + 36 = 52\) and \((6\sqrt{2})^2 = 72\). Hence the Pythagorean theorem does not hold as 52 doesn’t equal 72 (52 ≠ 72).

Step 3: Conclusion

Since the Pythagorean theorem doesn't hold, these points do not form a right triangle.

Key Concepts

The Pythagorean Theorem

The Pythagorean theorem is a fundamental principle in geometry, primarily dealing with right triangles. According to this theorem, 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. This can be expressed mathematically as
\[ a^2 + b^2 = c^2 \]
where \(a\) and \(b\) are the lengths of the legs and \(c\) is the length of the hypotenuse. This theorem is not only a staple in geometry but is also used in various fields such as physics, engineering, and computer science.
To better understand, imagine a triangle with side lengths of 3 units, 4 units, and 5 units. Applying the theorem yields the following:
\[ 3^2 + 4^2 = 5^2 \] \[ 9 + 16 = 25 \]
Since the equation holds true, we can conclude that the triangle is indeed a right triangle. This theorem becomes essential to determine if a set of three points in a plane can form a right triangle, as shown in the exercise provided.

Distance Formula

When dealing with geometry on the coordinate plane, the distance formula is vital for calculating the precise distance between two points. This formula is derived from the Pythagorean theorem and is expressed as:
\[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]
Here, \(d\) represents the distance between the points, and \((x_1, y_1)\) and \((x_2, y_2)\) are the coordinates of the two points. For example, if you are asked to find the distance between the points (1, 2) and (4, 6), using the distance formula produces:
\[ d = \sqrt{(4 - 1)^2 + (6 - 2)^2} \] \[ d = \sqrt{3^2 + 4^2} \] \[ d = \sqrt{9 + 16} \] \[ d = \sqrt{25} \] \[ d = 5 \]
The distance between these two points is 5 units. Understanding how to apply this formula correctly helps in solving problems involving lengths and distances on the coordinate plane, providing a crucial link between algebraic calculations and geometric interpretations.

Geometry in the Coordinate Plane

Geometry in the coordinate plane refers to the study of shapes and their relationships when plotted on a graph with perpendicular axes, usually labeled as the x-axis and y-axis. This allows for a visual representation of geometric figures and an analytical approach to solving problems using algebra.
In the context of right triangles, once the vertices are plotted on the coordinate plane, one can utilize the distance formula to find the lengths of the sides and then apply the Pythagorean theorem to determine if the triangle is right-angled. This combination of algebra and geometry enhances problem-solving skills and aids in visualizing complex concepts.
This approach was showcased in our exercise where we verified whether a set of points formed a right triangle by calculating distances between the points and testing these distances against the Pythagorean theorem. The result was a clear demonstration of the interplay between algebraic formulas and geometric principles to solve coordinate geometry problems.