Problem 58 - Algebra A Combined Function

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, as sides BC and AC are equal.

Step 1: Identify the vertices

The vertices of the triangle are given as \((2,6), (0,-2),\) and \((5,1)\). Label them as \(A(2,6), B(0,-2),\) and \(C(5,1)\).

Step 2: Calculate the length of AB

Use the distance formula to find the length between points \(A\) and \(B\): \[AB = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} = \sqrt{(0 - 2)^2 + (-2 - 6)^2} = \sqrt{4 + 64} = \sqrt{68}\].

Step 3: Calculate the length of BC

Use the distance formula to find the length between points \(B\) and \(C\): \[BC = \sqrt{(x_3 - x_2)^2 + (y_3 - y_2)^2} = \sqrt{(5 - 0)^2 + (1 - (-2))^2} = \sqrt{25 + 9} = \sqrt{34}\].

Step 4: Calculate the length of AC

Use the distance formula to find the length between points \(A\) and \(C\): \[AC = \sqrt{(x_3 - x_1)^2 + (y_3 - y_1)^2} = \sqrt{(5 - 2)^2 + (1 - 6)^2} = \sqrt{9 + 25} = \sqrt{34}\].

Step 5: Compare the side lengths

The lengths calculated are \(AB = \sqrt{68}\), \(BC = \sqrt{34}\), and \(AC = \sqrt{34}\). Since \(BC = AC\), two sides are equal.

Step 6: Determine triangle type

Since two sides of the triangle \((BC\) and \(AC)\) are equal, the triangle is isosceles.

Key Concepts

Distance Formula

The distance formula is a crucial tool in geometry for calculating the length between two points in a coordinate plane. Using the formula, you can easily find how far apart any two points are. It's very useful for problems involving triangles, like determining whether a triangle is isosceles.

To apply the distance formula between two points \(x_1, y_1\) and \(x_2, y_2\), the calculation is:

• Start with the x-coordinates: \(x_2 - x_1\)
• Square this difference.
• Do the same for the y-coordinates: \(y_2 - y_1\).
• Square this difference as well.
• Add up the squared differences: \( (x_2 - x_1)^2 + (y_2 - y_1)^2 \).
• Finally, take the square root of the sum to get the distance.

For instance, when you have points like \( (2, 6) \) and \( (0, -2) \), as presented in this problem, simply plug them into the formula to find the distance. It's straightforward once you break it down step by step.

Triangle Vertices

Vertices of a triangle are the corner points where two sides meet, and each vertex is defined by its coordinates in the plane. In our problem, the triangle has vertices \( (2, 6), (0, -2) \), and \( (5, 1) \).

Understanding the vertex concept is important to set the stage for solving geometry problems. Here’s why:

• Each vertex connects two sides of a triangle.
• You label vertices as \(A, B, C\) typically, making it easier to refer to them.
• Identifying the correct coordinates for each vertex is the first step for many calculations, including distance and angles.

By using the coordinates of vertices, you can calculate side lengths with the distance formula, which is then used to determine properties like the type of triangle you have.

Isosceles Triangle Calculation

An isosceles triangle is one where at least two sides have the same length. This property makes isosceles triangles special, as they have unique properties and are a common feature in many geometry problems.

To determine if a triangle is isosceles from coordinates, you:

• First, use the distance formula to find the lengths of all sides.
• Compare the lengths of these sides.
• If at least two sides are the same, the triangle is isosceles.

In this exercise, we calculated sides \(AB = \sqrt{68}\), \(BC = \sqrt{34}\), and \(AC = \sqrt{34}\). Since \(BC\) and \(AC\) are equal, this confirms that the triangle is isosceles.

Understanding this concept is key, because recognizing an isosceles triangle can lead you to other conclusions and simplify your problem-solving process.