Problem 86 - Algebra A Combined Function

Question

Show that a triangle with vertices at the points \((1,1),(-4,4),\) and (-3,0) is a right triangle.

Step-by-Step Solution

Answer

The triangle is a right triangle since it satisfies the Pythagorean theorem.

Step 1: Identify Points and Formulate Sides

The given vertices of the triangle are \((1,1), (-4,4),\) and \((-3,0)\). Let's denote these points as \(A = (1,1), B = (-4,4), C = (-3,0)\). To check if this triangle is right, we need to find the lengths of sides \(AB, BC,\) and \(CA\) using the distance formula.

Step 2: Calculate Length of Side AB

The length of side \(AB\) is calculated using the distance formula: \[AB = \sqrt{((-4) - 1)^2 + (4 - 1)^2} = \sqrt{(-5)^2 + 3^2} = \sqrt{25 + 9} = \sqrt{34}\]

Step 3: Calculate Length of Side BC

The length of side \(BC\) is calculated as follows: \[BC = \sqrt{((-3) + 4)^2 + (0 - 4)^2} = \sqrt{1^2 + (-4)^2} = \sqrt{1 + 16} = \sqrt{17}\]

Step 4: Calculate Length of Side CA

The length of side \(CA\) can be calculated using the formula: \[CA = \sqrt{((-3) - 1)^2 + (0 - 1)^2} = \sqrt{(-4)^2 + (-1)^2} = \sqrt{16 + 1} = \sqrt{17}\]

Step 5: Verify Right Triangle Condition

For the triangle to be right-angled, the Pythagorean theorem must hold for the three sides. Check if any pair of the squared side lengths adds up to the third squared side's length.Compare: - \(AB^2 + BC^2 = \sqrt{34}^2 + \sqrt{17}^2 = 34 + 17 = 51\) - \(AB^2 + CA^2 = 34 + 17 = 51\)- \(BC^2 + CA^2 = \sqrt{17}^2 + \sqrt{17}^2 = 17 + 17 = 34\)Since \(BC^2 + CA^2 = AB^2\), the triangle satisfies the Pythagorean theorem, and hence it is a right triangle.

Key Concepts

Distance Formula

The distance formula is an essential tool in coordinate geometry. It allows us to calculate the distance between any two points in a plane. By using the coordinates of these points, we can derive the length of the line segment that connects them.
To apply the formula, you need coordinates of two points, say \((x_1, y_1)\) and \((x_2, y_2)\). The distance \(d\) between these points can be found using:

• \(d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}\).

The formula stems from the concept of computing the length of the hypotenuse in a right triangle, formed by the projections of the line segment onto the coordinate axes. This makes it similar to the Pythagorean theorem, where we square the differences in the \(x\) and \(y\) coordinates, reflecting the horizontal and vertical changes. Summing these squares gives us the square of the diagonal (or actual distance), and taking the square root provides the distance itself.
This formula is a powerful way to bridge geometry and algebra, showing how spatial relationships can be represented and calculated using numbers.

Pythagorean Theorem

The Pythagorean Theorem is a fundamental principle in geometry. It applies to right triangles, where it relates the lengths of the sides. The theorem states:

• In a right triangle, the square of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides.

Mathematically, if a triangle has sides \(a\), \(b\), and hypotenuse \(c\), then:

• \(c^2 = a^2 + b^2\)

To understand why the triangle from our example is right-angled, we apply this theorem. We calculated the squares of all sides and observed that the sum of the squares of two sides equals the square of the third side. This confirmation solidifies that one of the angles is a right angle.
The Pythagorean theorem is not only used in geometry but also serves as a base for many proofs and applications in various mathematical disciplines.

Coordinate Geometry

Coordinate geometry, also known as analytic geometry, allows us to use algebra to handle geometric problems. By placing geometric figures on the plane using coordinates, we can explore their properties more algebraically.
In the coordinate plane, every point is identified by an ordered pair of numbers (or coordinates), typically written as \((x, y)\). This branches out into understanding relationships between points, lines, and shapes using algebraic equations and expressions.

• Understanding the slope of a line between two points: \(m = \frac{y_2 - y_1}{x_2 - x_1}\)
• Finding equations of lines: point-slope form, slope-intercept form, etc.
• Using intersection points to solve geometric problems.

In the right triangle problem, coordinate geometry uses the distance formula to compute sides' lengths, which leads to verifying the Pythagorean theorem. This synergy between algebra and geometry enhances problem-solving skills, giving a structured way to approach spatial relationships.