Problem 96 - Intermediate Algebra

Question

Determine whether the triangle with vertices \((2,6),(0,-2)\) and \((5,1)\) is an isosceles triangle.

Step-by-Step Solution

Answer

The triangle is isosceles with two sides of length \( \sqrt{34} \).

Step 1: Find Distance Between Points

Use the distance formula \( d = \sqrt{(x_2-x_1)^2 + (y_2-y_1)^2} \) to calculate the distance between each pair of points. For points \((2,6)\) and \((0,-2)\): \( d_1 = \sqrt{(0-2)^2 + (-2-6)^2} = \sqrt{4 + 64} = \sqrt{68} \).For points \((0,-2)\) and \((5,1)\): \( d_2 = \sqrt{(5-0)^2 + (1+2)^2} = \sqrt{25 + 9} = \sqrt{34} \).For points \((2,6)\) and \((5,1)\): \( d_3 = \sqrt{(5-2)^2 + (1-6)^2} = \sqrt{9 + 25} = \sqrt{34} \).

Step 2: Compare the Distances

An isosceles triangle has at least two sides of equal length. Compare the calculated distances: - \( d_1 = \sqrt{68} \)- \( d_2 = \sqrt{34} \)- \( d_3 = \sqrt{34} \)We see that \( d_2 = d_3 \), therefore, the triangle has two sides of equal length.

Step 3: State Conclusion

Since the triangle has two sides of equal length (\( d_2 = d_3 \)), it is an isosceles triangle.

Key Concepts

Distance Formula

The distance formula is a vital tool in geometry, particularly in coordinate geometry. It's like a magic formula that helps you calculate the distance between two points on a coordinate plane. Imagine you have two points, \(x_1, y_1\) and \(x_2, y_2\). The distance between them can be found using the formula:
\[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \].
This formula is derived from the Pythagorean Theorem and gives you a way to "walk diagonally" across the coordinate plane, measuring the line segment that connects the two points.

• First, you subtract the x-coordinates and square the result to find the horizontal distance.
• Then, you subtract the y-coordinates and square that result to find the vertical distance.
• Finally, you add these squares together and take the square root to get the total distance.

This method allows you to determine distances accurately and is essential for understanding how points relate to each other in a plane. Once you grasp this concept, solving problems like finding the lengths of sides in triangles becomes straightforward.

Coordinate Geometry

Coordinate geometry, also known as analytic geometry, is where algebra meets geometry. It allows us to study geometric figures by placing them on a coordinate plane, using equations and coordinates. This field revolutionizes the way we analyze geometric shapes by using algebraic formulas.
For instance, imagine plotting points on a grid, like crafting a picture through dots. Each point, defined by an \(x, y\) coordinate, simplifies calculations and visualizations.
Here are a few key tasks you can perform with coordinate geometry:

• Identify the exact location of points using their coordinates.
• Calculate distances between points using the distance formula.
• Classify triangles by their sides and angles, as we do with isosceles triangles.
• Find midpoints, slopes, and line equations.

Coordinate geometry opens a door to a more analytical approach to geometry, making it possible to apply algebraic techniques to solve and understand geometric problems. Understanding it is crucial for broader mathematical problem-solving.

Triangle Classification

Triangles come in various types, primarily classified based on their sides and angles. This classification helps in understanding the properties and characteristics of a triangle.

Classification by Sides

There are three primary types of triangles based on their sides:

• Equilateral Triangle: All sides are equal, and consequently, all angles are equal, each being 60 degrees.
• Isosceles Triangle: Has at least two sides of equal length. As seen in the exercise, if two distances between points are the same, it means the triangle is isosceles.
• Scalene Triangle: All sides have different lengths, and thus, all angles are different.

Classification by Angles

Triangles can also be classified by their angle sizes:

• Acute Triangle: All angles are less than 90 degrees.
• Right Triangle: One angle is exactly 90 degrees.
• Obtuse Triangle: One angle is more than 90 degrees.

Knowing these classifications is fundamental for understanding triangles and solving geometrical problems. It also helps in identifying triangle properties, such as congruence and similarity, essential concepts across various fields of math.