Problem 37 - Precalculus Mathematics for Calculus

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 CA are equal in length.

Step 1: Calculate the Length of Side AB

Use the distance formula between points \(A(0, 2)\) and \(B(-3, -1)\). The formula for the distance between two points \((x_1, y_1)\) and \((x_2, y_2)\) is: \(\sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}\). For points \(A\) and \(B\), it becomes: \[\sqrt{(-3 - 0)^2 + (-1 - 2)^2} = \sqrt{9 + 9} = \sqrt{18} = 3\sqrt{2}.\]

Step 2: Calculate the Length of Side BC

Use the distance formula between points \(B(-3, -1)\) and \(C(-4, 3)\). Apply the distance formula: \[\sqrt{(-4 - (-3))^2 + (3 - (-1))^2} = \sqrt{(-1)^2 + 4^2} = \sqrt{1 + 16} = \sqrt{17}.\]

Step 3: Calculate the Length of Side CA

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

Step 4: Compare the Side Lengths

The lengths of the sides are \(AB = 3\sqrt{2}\), \(BC = \sqrt{17}\), and \(CA = \sqrt{17}\). Notice that \(BC = CA\). This shows that two sides of the triangle are equal.

Key Concepts

Distance Formula

The distance formula is a powerful tool in coordinate geometry for finding the distance between two points in a plane. It is derived from the Pythagorean theorem and provides an exact measure of the straight-line distance between two points with given coordinates. In practice, if you have two points

• Point 1 with coordinates \((x_1, y_1)\)
• Point 2 with coordinates \((x_2, y_2)\)

you can determine the distance between them using the formula:\[\sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}\]This formula calculates the horizontal and vertical differences between the points first (\((x_2 - x_1)\) and \((y_2 - y_1)\)) and then applies the Pythagorean theorem to find the hypotenuse, which is the distance between the points. In the context of our triangle:

• The distance between points A and B: \(\sqrt{18} = 3\sqrt{2}\)
• The distance between points B and C: \(\sqrt{17}\)
• The distance between points C and A: \(\sqrt{17}\)

These calculations enable us to explore the properties of the triangle more effectively.

Coordinate Geometry

Coordinate geometry, also known as analytical geometry, uses a coordinate system to investigate geometric properties. It's a branch of geometry where algebraic techniques are employed to solve geometric problems. This approach allows us to work with complex shapes more conveniently by using numbers and equations rather than abstract shapes.In the case of the triangle with vertices at \(A(0,2)\), \(B(-3,-1)\), and \(C(-4,3)\),coordinate geometry helps us determine the lengths of the sides accurately using the distance formula. By knowing the coordinates of points:

• We can easily calculate distances between any two points
• Visualize the shape and size of the triangles
• And verify relationships, like parallelism or perpendicularity, based on slopes

Coordinate geometry also helps establish whether the triangle is isosceles by computing the lengths of sides based on their coordinates. This method offers precision, which is often difficult to achieve with simple visual inspection or traditional geometric methods.

Triangle Properties

Triangles, simple as they may seem, have rich properties and classifications. Understanding the types and properties of triangles is essential in geometry. A key property is that isosceles triangles have at least two sides of equal length.In an isosceles triangle:

• Two sides are equal in length
• Two angles opposite these equal sides are also equal

This exercise explores whether the triangle formed by points A, B, and C is isosceles, based on the comparison of side lengths calculated using the distance formula:

• Sides AB, BC, and CA were calculated as \(3\sqrt{2}, \sqrt{17}, \text{and } \sqrt{17}\)
• Since sides BC and CA are both \(\sqrt{17}\), they are equal
• Thus, the triangle is isosceles because it fulfills the condition of having two equal sides

By understanding these concepts, students can easily identify and classify triangles based on side lengths and angles. Recognizing an isosceles triangle when given coordinates becomes straightforward with these methods, demonstrating the strength of combining algebra with geometry.