Question

Triangle \(F G H\) has vertices \(F(2,4), G(0,2)\), and \(H(3,-1)\). Determine whether \(\triangle F G H\) is a right triangle. Explain.

Short answer

Triangle FGH is a right triangle by the Pythagorean theorem.

Step-by-step solution

Calculate the lengths of the sides

First, we'll calculate the length of each side of the triangle using the distance formula: \[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}\]1. For side FG: \[d_{FG} = \sqrt{(0 - 2)^2 + (2 - 4)^2} = \sqrt{4 + 4} = \sqrt{8} = 2\sqrt{2}\]2. For side GH: \[d_{GH} = \sqrt{(3 - 0)^2 + (-1 - 2)^2} = \sqrt{9 + 9} = \sqrt{18} = 3\sqrt{2}\]3. For side HF: \[d_{HF} = \sqrt{(3 - 2)^2 + (-1 - 4)^2} = \sqrt{1 + 25} = \sqrt{26}\]

Verify the Pythagorean theorem

We use the Pythagorean theorem to determine if \(\triangle FGH\) is a right triangle. The theorem states that in a right triangle, the square of the length of the hypotenuse is equal to the sum of the squares of the lengths of the other two sides.Assume \(HF\) is the hypotenuse, and verify:\[(2\sqrt{2})^2 + (3\sqrt{2})^2 = (\sqrt{26})^2\]- Calculate the left side: - \((2\sqrt{2})^2 = 8\) - \((3\sqrt{2})^2 = 18\) - Sum: \(8 + 18 = 26\)- Calculate the right side: - \((\sqrt{26})^2 = 26\)- Since both sides are equal, \(26 = 26\), \(\triangle FGH\) is a right triangle.

Key concepts

Distance Formula

The distance formula is a crucial tool in geometry. It helps us find the straight-line distance between two points in a coordinate plane. This formula originates from the Pythagorean theorem and calculates distance by considering the horizontal and vertical differences between two points.

To apply the distance formula, which is written as \[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}\]you should:

• Identify the coordinates of the two points. For example, point \(A(x_1, y_1)\) and point \(B(x_2, y_2)\).
• Subtract the x-coordinates: \(x_2 - x_1\).
• Subtract the y-coordinates: \(y_2 - y_1\).
• Square both results and add them together.
• Take the square root of the sum to find the distance.

Understanding this formula helps with solving a variety of problems involving triangles and other geometric shapes. Knowing how distances between points are calculated can also assist in navigational problems and engineering solutions.

Pythagorean Theorem

The Pythagorean theorem is one of the foundations of geometric studies, particularly involving right triangles. It states that 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.

The theorem is expressed as:
\[ c^2 = a^2 + b^2 \]
where \(c\) represents the hypotenuse, and \(a\) and \(b\) are the other two sides. This relation only holds true for right triangles, making it a valuable theorem for checking whether a triangle is right-angled.

When applying it to a problem:

• First, identify which side could potentially be the hypotenuse by finding the longest side.
• Use the formula to calculate whether the square of this side is indeed the sum of the squares of the other two sides.
• If the equation holds true, you confirmed it's a right triangle.

This theorem plays a crucial role in geometry, trigonometry, and even in diverse practical fields such as construction and computer graphics.

Coordinate Geometry

Coordinate geometry, also known as analytic geometry, is a branch of mathematics that uses algebraic equations to describe geometric shapes. This approach allows us to study geometry through a coordinate system, like the Cartesian plane.

In coordinate geometry:

• We use coordinates (x, y) to represent points.
• Equations represent lines, curves, and other figures.
• We apply algebraic techniques to determine lengths, midpoints, slopes, and other properties of geometric figures.

For example, when analyzing triangles in a coordinate plane:

• Calculate distances using the distance formula.
• Apply the Pythagorean theorem to verify right angles.
• Use points to determine intersections and angles.

This field provides a more visual representation of algebraic problems and enables complex problem-solving through graphical interpretations. Understanding coordinate geometry is essential not only for academic pursuits but also for practical applications such as mapping and navigation.