Problem 43 - Algebra and Trigonometry

Question

Show that the triangle with vertices \(A(0,2), B(-3,-1),\) and \(C(-4,3)\) is isosceles.

Step-by-Step Solution

Answer

Triangle ABC is isosceles because sides BC and AC are equal.

Step 1: Calculate Distance AB

Use the distance formula to find the length of side AB. The distance formula is \(d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}\). For points \(A(0,2)\) and \(B(-3,-1)\), compute: \( AB = \sqrt{(-3-0)^2 + (-1-2)^2} = \sqrt{9 + 9} = \sqrt{18} = 3\sqrt{2} \).

Step 2: Calculate Distance BC

Next, find the length of side BC using the distance formula with points \(B(-3,-1)\) and \(C(-4,3)\). Compute: \( BC = \sqrt{(-4 + 3)^2 + (3 + 1)^2} = \sqrt{1 + 16} = \sqrt{17} \).

Step 3: Calculate Distance AC

Find the length of side AC using the distance formula for points \(A(0,2)\) and \(C(-4,3)\). Compute: \( AC = \sqrt{(-4 - 0)^2 + (3 - 2)^2} = \sqrt{16 + 1} = \sqrt{17} \).

Step 4: Compare Side Lengths

Compare the distances calculated: \( AB = 3\sqrt{2} \), \( BC = \sqrt{17} \), and \( AC = \sqrt{17} \). Since \( BC = AC \), we conclude that triangle ABC has two sides of equal length, identifying it as an isosceles triangle.

Key Concepts

Distance Formula

The distance formula is a vital tool in coordinate geometry that helps you calculate the distance between two points on a plane. This formula is the key to determining if a triangle is isosceles. To use it, apply the formula:

• Identify your two points, labeled \((x_1, y_1)\) and \((x_2, y_2)\).
• Insert these coordinates into the equation: \[d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}\].

This simple formula relies on the Pythagorean theorem, breaking down the calculation of a distance into horizontal and vertical components. By squaring the differences and adding them, you essentially calculate the hypotenuse of a right triangle formed by these two points.
In practice, as seen in the given problem, applying the distance formula to each pair of points allows for the direct comparison of side lengths to check for isosceles properties.

Triangle Vertices

Understanding vertices is crucial in the study of triangles, particularly within coordinate geometry. The vertices of a triangle are simply its corner points. Each vertex is represented as a pair of coordinates. For instance, in the triangle ABC from the exercise, the vertices are labeled as points A \((0,2)\), B \((-3,-1)\), and C \((-4,3)\).
Vertices are foundational because they provide the necessary data to calculate side lengths. Simply plug each set of coordinates into the distance formula to find the side lengths between each pair of points.
With this information, it becomes easy to determine the type of triangle, such as checking for equal-length sides which characterize isosceles triangles.

Coordinate Geometry

Coordinate geometry, or analytic geometry, is the study of geometric figures using a coordinate system. This field allows the exploration of geometric concepts like distances, angles, and slopes through algebraic equations.
By placing points in a coordinate plane, you can easily compute attributes of shapes. The exercise involving the three points A, B, and C exemplifies how coordinate geometry enables the analysis of a triangle's properties.

• Vertices are plotted based on their coordinates.
• Distances between vertices are calculated using the distance formula.

This analytical approach helps navigate complex shapes and determine congruency, leading to classifications like isosceles.
Thus, coordinate geometry is a powerful tool for verifying geometric properties through clear and systematic calculations.