Problem 37 - 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

The triangle is isosceles because sides BC and AC are equal.

Step 1: Find the Distance Between Points A and B

The distance between two points \( (x_1, y_1) \) and \( (x_2, y_2) \) can be found using the distance formula: \(D = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}\)Calculating \(AB\), we have:\[AB = \sqrt{(-3 - 0)^2 + (-1 - 2)^2} = \sqrt{(-3)^2 + (-3)^2} = \sqrt{9 + 9} = \sqrt{18} = 3\sqrt{2}\]

Step 2: Find the Distance Between Points B and C

Using the distance formula for the points \(B(-3, -1)\) and \(C(-4, 3)\):\[BC = \sqrt{(-4 - (-3))^2 + (3 - (-1))^2} = \sqrt{(-1)^2 + (4)^2} = \sqrt{1 + 16} = \sqrt{17}\]

Step 3: Find the Distance Between Points A and C

Using the distance formula for the points \(A(0, 2)\) and \(C(-4, 3)\):\[AC = \sqrt{(-4 - 0)^2 + (3 - 2)^2} = \sqrt{(-4)^2 + (1)^2} = \sqrt{16 + 1} = \sqrt{17}\]

Step 4: Compare the Distances

We have calculated the sides:\[AB = 3\sqrt{2}, \quad BC = \sqrt{17}, \quad AC = \sqrt{17}\]Since \(BC = AC\), two sides of the triangle are equal. Thus, the triangle is isosceles.

Key Concepts

Distance Formula

The distance formula is a fundamental tool in coordinate geometry. It allows us to calculate the distance between two points on a plane, using their coordinates. This is essential when working with geometric shapes, like triangles. To use the distance formula, you take two points, say \( (x_1, y_1) \) and \( (x_2, y_2) \), and plug them into the formula: \[D = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}\]The process involves:

• Subtracting the x-coordinates \(x_2 - x_1\) and the y-coordinates \(y_2 - y_1\).
• Squaring each of these differences.
• Adding the squared numbers together.
• Finally, taking the square root of this sum gives the distance \(D\).

In our exercise, this method was applied between different pairs of points to find the sides of the triangle \(AB\), \(BC\), and \(AC\). Each distance calculation confirms the length of the sides, which is crucial to determining if a triangle is isosceles.

Coordinate Geometry

Coordinate geometry, also known as analytic geometry, is a branch of mathematics that uses a coordinate system to investigate geometric properties. It combines algebra and geometry, allowing for a detailed study of shapes, sizes, and relative positions of figures.
The basis of coordinate geometry is the coordinate plane, consisting of the x-axis and y-axis, which intersect at the origin (0,0). This framework makes it possible to represent geometric objects, such as points, lines, and curves, with algebraic equations.
In the context of our exercise, coordinate geometry was employed to locate the vertices of the triangle in the plane. The points \(A(0,2)\), \(B(-3,-1)\), and \(C(-4,3)\) were used to set up the triangle and then to explore the triangle's properties using the distance formula.

Triangle Properties

Triangles are one of the basic shapes in geometry with several important properties. Understanding these properties can help you identify different types of triangles, such as isosceles, equilateral, and scalene.
An isosceles triangle is characterized by having at least two sides of equal length, which also means two angles will be equal since each angle is opposite a side. The recognition that a triangle is isosceles can be very useful in solving related problems such as finding missing angles or calculating perimeters.

• If two sides are the same length, the angles opposite those sides are equal.
• The altitude drawn from the apex of the isosceles triangle to the base bisects the base.
• The triangle's symmetry makes calculations and deductions simpler.

In the exercise, the triangle with vertices \(A, B,\) and \(C\) was shown to be isosceles by demonstrating that \(BC = AC\), meaning two sides are equal. This was a direct consequence of using the distance formula effectively.