Problem 13 - Precalculus

Question

In each case, determine whether the triangle with the given vertices is a right triangle. (a) (7,-1),(-3,5),(-12,-10) (b) (4,5),(-3,9),(1,3) (c) (-8,-2),(1,-1),(10,19)

Step-by-Step Solution

Answer

(a) Yes, it's a right triangle. (b) No, it's not a right triangle. (c) No, it's not a right triangle.

Step 1: Recall the Definition of a Right Triangle

A right triangle is one with one angle measuring exactly 90 degrees. To determine if a triangle is right, we can use the distance formula to find the lengths of the sides, and then apply the Pythagorean Theorem: If \( c^2 = a^2 + b^2 \), where \( c \) is the longest side, the triangle is a right triangle.

Step 2: Calculate the Distances Between Points (a)

To find if the triangle with vertices (7,-1),(-3,5), and (-12,-10) is right, calculate the distances:1. Between (7,-1) and (-3,5): \[ d_1 = \sqrt{(-3 - 7)^2 + (5 - (-1))^2} = \sqrt{100 + 36} = 2\sqrt{34} \]2. Between (7,-1) and (-12,-10): \[ d_2 = \sqrt{(-12 - 7)^2 + (-10 - (-1))^2} = \sqrt{361 + 81} = 20 \]3. Between (-3,5) and (-12,-10): \[ d_3 = \sqrt{(-12 + 3)^2 + (-10 - 5)^2} = \sqrt{81 + 225} = 18 \]

Step 3: Check the Pythagorean Theorem for Case (a)

For the triangle, check if the longest side's square equals the sum of the squares of the other two sides:\[ 20^2 = (2\sqrt{34})^2 + 18^2 \]\[ 400 = 136 + 324 \]Since 400 = 400, the triangle is a right triangle.

Step 4: Calculate the Distances Between Points (b)

For points (4,5),(-3,9),(1,3), calculate:1. Between (4,5) and (-3,9): \[ d_1 = \sqrt{(-3 - 4)^2 + (9 - 5)^2} = \sqrt{49 + 16} = \sqrt{65} \]2. Between (4,5) and (1,3): \[ d_2 = \sqrt{(1 - 4)^2 + (3 - 5)^2} = \sqrt{9 + 4} = \sqrt{13} \]3. Between (-3,9) and (1,3): \[ d_3 = \sqrt{(1 + 3)^2 + (3 - 9)^2} = \sqrt{16 + 36} = \sqrt{52} \]

Step 5: Check the Pythagorean Theorem for Case (b)

For the triangle, verify:\[ \sqrt{65}^2 = \sqrt{13}^2 + \sqrt{52}^2 \]\[ 65 eq 13 + 52 \]Since the squared sum does not match, the triangle is not a right triangle.

Step 6: Calculate the Distances Between Points (c)

For points (-8,-2),(1,-1),(10,19), calculate:1. Between (-8,-2) and (1,-1): \[ d_1 = \sqrt{(1 + 8)^2 + (-1 + 2)^2} = \sqrt{81 + 1} = \sqrt{82} \]2. Between (-8,-2) and (10,19): \[ d_2 = \sqrt{(10 + 8)^2 + (19 + 2)^2} = \sqrt{324 + 441} = 27 \]3. Between (1,-1) and (10,19): \[ d_3 = \sqrt{(10 - 1)^2 + (19 + 1)^2} = \sqrt{81 + 400} = 21 \]

Step 7: Check the Pythagorean Theorem for Case (c)

For this triangle, verify if:\[ 27^2 = \sqrt{82}^2 + 21^2 \]\[ 729 = 82 + 441 \]Since 729 = 523, this triangle is also not a right triangle.

Key Concepts

Distance Formula

To determine if a triangle is a right triangle using coordinates, we first calculate the distances between the points using the distance formula. This helps us find the length of each side of the triangle, which is crucial for further calculations. The distance formula is given by:
\[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]
where \((x_1, y_1)\) and \((x_2, y_2)\) are the coordinates of two points. This formula gives us the Euclidean distance, which is essentially the length of the line segment joining these two points.

• Use the formula for each side by pairing points.
• Plug the values from the points into the formula to compute the length of each side.
• Make sure to take the square root of the sum for the correct distance.

Understanding the distance formula is essential as it sets the stage for checking if the triangle is right by comparing the lengths calculated.

Pythagorean Theorem

Once the lengths of the triangle's sides are known using the distance formula, the Pythagorean Theorem comes into play to determine if the triangle is a right triangle. According to the Pythagorean Theorem:
\[ c^2 = a^2 + b^2 \]
where \(c\) is the longest side (hypotenuse) and \(a\) and \(b\) are the other two sides. The theorem is applicable if one of the angles is exactly 90 degrees.

• Identify the longest side, which serves as the potential hypotenuse.
• Square each side's length and check if the sum of the squares of the two shorter sides equals the square of the longest side.
• If the relationship holds true, the triangle is a right triangle; otherwise, it isn't.

This theorem is a fundamental principle in geometry and confirms whether a triangle has a right angle by using side lengths, bridging the basic algebra with geometric proof.

Triangle Geometry

In triangle geometry, understanding the configuration of a triangle based on its vertices and side lengths is vital. A right triangle has unique properties, specifically having one 90-degree angle. The properties of a right triangle manifest through their side lengths and can be easily identified using geometric methods, such as calculating angles or confirming the Pythagorean identity.

• Right triangles have one 90-degree angle.
• The longest side opposite the right angle is the hypotenuse.
• Known theorems and distance calculations assist in identifying the right-angle property.

Grasping these geometric principles is important as it flows naturally to using mathematical techniques, like formulas, to derive these configurations precisely.

Coordinates of Vertices

When presented with the vertices of a triangle, such as points like \((x_1, y_1)\), \((x_2, y_2)\), and \((x_3, y_3)\), the coordinate plane allows us to visualize and compute efficiently. Coordinates serve as the foundation for applying geometric concepts and offer a graphical representation of the triangle. Here's how they come into play:

• Identify the position of each vertex and how they form the triangle.
• Calculate distances between each pair of vertices to establish side lengths.
• Using these coordinates, the triangle's spatial arrangement aids in further analysis.

Coordinates are fundamental because they provide a blueprint to calculate other important aspects, like side lengths and angles, using geometry within the coordinate plane.