Problem 28 - Contemporary Precalculus

Question

Show that the three points are the vertices of a right triangle, and state the length of the hypotenuse. I You may assume that a triangle with sides of lengths \(a, b, c\) is a right triangle with hypotenuse c provided that \(a^{2}+b^{2}=c^{2} .1\). $$(3,-2),(0,4),(-2,3)$$

Step-by-Step Solution

Answer

If so, what is the length of the hypotenuse? Answer: Yes, the points form a right triangle, and the length of the hypotenuse is \(5\sqrt{2}\).

Step 1: Find the distances between the points

First, we need to find the distances between the given points using the distance formula: Distance between points \((3,-2)\) and \((0,4)\): $$d_{(3,-2),(0,4)}=\sqrt{(0-3)^2+(4-(-2))^2}=\sqrt{9+36}=3\sqrt{5}$$ Distance between points \((3,-2)\) and \((-2,3)\): $$d_{(3,-2),(-2,3)}=\sqrt{(-2-3)^2+(3-(-2))^2}=\sqrt{25+25}=5\sqrt{2}$$ Distance between points \((0,4)\) and \((-2,3)\): $$d_{(0,4),(-2,3)}=\sqrt{(-2-0)^2+(3-4)^2}=\sqrt{4+1}= \sqrt{5}$$ The distances are \(3\sqrt{5}\), \(5\sqrt{2}\), and \(\sqrt{5}\).

Step 2: Check if the points form a right triangle

Now we need to check if the square of the lengths of the sides fulfills the Pythagorean theorem (\(a^2+b^2=c^2\)): We have the squares of the lengths: - \((3\sqrt{5})^2 = 45\) - \((5\sqrt{2})^2 = 50\) - \((\sqrt{5})^2 = 5\) Let's check if we can find a combination where \(a^2+b^2=c^2\): \(45 + 5 = 50\) is a valid combination that fulfills the Pythagorean theorem.

Step 3: Determine the hypotenuse

Since \((3\sqrt{5})^2\) and \((\sqrt{5})^2\) are the smaller squares whose sum is equal to \((5\sqrt{2})^2\), the side with length \(5\sqrt{2}\) is the hypotenuse of the right triangle. Therefore, the three points indeed form a right triangle with a hypotenuse of length \(5\sqrt{2}\).

Key Concepts

Right Triangle

A right triangle is a special type of triangle that has one angle equal to 90 degrees. This angle is known as the right angle. What makes the right triangle unique is the relationship between its sides. The longest side, opposite the right angle, is called the hypotenuse, while the other two sides are referred to as the legs.

In any right triangle, there's a geometric connection that holds true, known as the Pythagorean Theorem. This theorem helps us to identify if a triangle is indeed a right triangle by looking at the lengths of its sides.

• Right triangles are often used in various geometry problems because of their predictable properties.
• They serve as a foundation for trigonometry, which deals with angles and sides of triangles.

Understanding the structure of right triangles is essential in solving many math problems, especially in coordinate geometry.

Pythagorean Theorem

The Pythagorean Theorem is a fundamental concept in geometry, especially when dealing with right triangles. It states that in a right triangle, the sum of the squares of the lengths of the two shorter sides (legs) is equal to the square of the length of the longest side (hypotenuse). This can be expressed mathematically as: \[a^2 + b^2 = c^2\] Here, \(a\) and \(b\) are the lengths of the legs and \(c\) is the length of the hypotenuse.

When given three side lengths, you can use the Pythagorean Theorem to determine if they form a right triangle:

• If the theorem holds true for the side lengths, then you have a right triangle.
• It provides a reliable method to check the authenticity of the triangle's right angle without having to measure the angle directly.

Applying this theorem allows us to explore many practical applications, both in mathematics and real-world situations, making it a powerful tool in problem-solving.

Coordinate Geometry

Coordinate geometry, also known as analytic geometry, involves using a coordinate system to study geometry. This approach combines algebra and geometry to handle problems involving geometric shapes in a more analytical manner.

Key concepts in coordinate geometry include:

• The Cartesian coordinate system, which locates points using ordered pairs of numbers (\(x, y\)).
• Distance formula for calculating the distance between two points. This helps in determining side lengths in geometry problems.

For instance, given two points \((x_1, y_1)\) and \((x_2, y_2)\), the distance between them is found using the formula: \[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \] In our triangle problem, we used this formula to find the distances between vertices to verify the condition of forming a right triangle. This illustrates the power of coordinate geometry in bridging the gap between algebraic calculations and geometric interpretations.