Problem 6 - Calculus Early Transcendentals: Pearson New International Edition

Question

Show that the triangle whose vertices are \((2,-4),(4,0)\), and \((8,-2)\) is a right triangle.

Step-by-Step Solution

Answer

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

Step 1: Find the Length of the Sides

To determine if the triangle is a right triangle, calculate the distance between each pair of vertices using the distance formula: \[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]- For vertices \((2, -4)\) and \((4, 0)\), the distance is \[ \sqrt{(4-2)^2 + (0+4)^2} = \sqrt{4 + 16} = \sqrt{20} = 2\sqrt{5} \]- For vertices \((4, 0)\) and \((8, -2)\), the distance is \[ \sqrt{(8-4)^2 + (-2-0)^2} = \sqrt{16 + 4} = \sqrt{20} = 2\sqrt{5} \]- For vertices \((2, -4)\) and \((8, -2)\), the distance is \[ \sqrt{(8-2)^2 + (-2+4)^2} = \sqrt{36 + 4} = \sqrt{40} = 2\sqrt{10} \]

Step 2: Check for the Pythagorean Theorem

In a right triangle, the square of the length of the hypotenuse must equal the sum of the squares of the lengths of the other two sides. Consider the lengths:- \(2\sqrt{5}\), \(2\sqrt{5}\), and \(2\sqrt{10}\).Calculate:\[ (2\sqrt{5})^2 + (2\sqrt{5})^2 = (2\sqrt{10})^2 \]- \( (2\sqrt{5})^2 = 4 \times 5 = 20 \)- \( (2\sqrt{5})^2 = 20 \)- \( (2\sqrt{10})^2 = 4 \times 10 = 40 \) - Total: \(20 + 20 = 40 \)Since \(40 = 40\), the triangle follows the Pythagorean theorem, confirming it is a right triangle.

Key Concepts

Distance Formula

The distance formula is a crucial tool in coordinate geometry, allowing us to find the distance between two points on a plane. It's essentially an application of the Pythagorean theorem in a coordinate plane setting. For any two points, say \((x_1, y_1)\) and \((x_2, y_2)\), the formula is given by: \[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \] This formula helps calculate the straight-line distance between the two points.

• Calculate the horizontal distance: \((x_2 - x_1)\)
• Calculate the vertical distance: \((y_2 - y_1)\)
• Square each of these distances to eliminate any negative values
• Add them together and take the square root to find the actual distance

This is exactly what was done for the triangle in the exercise, resulting in side lengths of \(2\sqrt{5}\), \(2\sqrt{5}\), and \(2\sqrt{10}\).
Understanding how to apply this formula is fundamental and will be a great help in any geometry task involving coordinates.

Pythagorean Theorem

The Pythagorean theorem is a fundamental principle in geometry, particularly useful for verifying the properties of a right triangle. It states that in a right triangle, the sum of the squares of the lengths of the two shorter sides, known as legs, equals the square of the length of the longest side, known as the hypotenuse. For sides \(a\), \(b\), and hypotenuse \(c\), it is expressed as: \[ a^2 + b^2 = c^2 \] In the provided exercise, we used the calculated side lengths \(2\sqrt{5}\) and \(2\sqrt{10}\) to check if they conform to this theorem.
We worked it out to show:

• \((2\sqrt{5})^2 = 20\)
• \((2\sqrt{5})^2 = 20\)
• \((2\sqrt{10})^2 = 40\)
• Confirming \(20 + 20 = 40\), matching with \(40\)

This confirms that the triangle with the given side lengths is indeed a right triangle, as it satisfies the Pythagorean theorem.

Coordinate Geometry

Coordinate geometry, or analytic geometry, is a branch of mathematics where we study geometric figures through coordinates and algebraic expressions. This allows us to combine algebra and geometry and provides a powerful context for working with shapes on a plane.
One of the great advantages of coordinate geometry is its ability to determine properties of shapes, such as distances, angles, and symmetries, using numerical methods. This makes it particularly suited for verifying geometric properties of figures like triangles.
In the case of the exercise, we used coordinate geometry to establish that the triangle is a right triangle by calculating distances between points and applying the Pythagorean theorem. By putting the triangle in the coordinate plane, we could accurately and efficiently derive our solution.

• Use of coordinates like \((2, -4)\), \((4,0)\), and \((8,-2)\)
• Application of distance formula for side lengths
• Use of Pythagorean theorem to confirm a right angle

Understanding and applying the principles of coordinate geometry is invaluable in solving complex geometric problems efficiently.