Problem 12 - Precalculus

Question

Use the distance formula to show that, in each case, the triangle with given vertices is an isosceles triangle. (a) (0,2),(7,4),(2,-5) (b) (-1,-8),(0,-1),(-4,-4) (c) (-7,4),(-3,10),(1,3)

Step-by-Step Solution

Answer

All triangles are isosceles: (a) sides \(\sqrt{53}\), \(\sqrt{53}\); (b) sides \(\sqrt{25}\), \(\sqrt{25}\); (c) sides \(\sqrt{65}\), \(\sqrt{65}\).

Step 1: Understanding the Distance Formula

The distance formula is used to find the distance between two points in a plane, given by \(d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}\). We will apply this formula to find the lengths of the sides of the triangles formed by each set of given vertices.

Step 2: Calculate Sides for Triangle (a)

For vertices (0,2), (7,4), and (2,-5), calculate the lengths of the sides: - Side 1: "\((0,2)\) to \((7,4)\), " Calculating, we get \(\sqrt{(7-0)^2 + (4-2)^2} = \sqrt{49 + 4} = \sqrt{53}\). - Side 2: "\((7,4)\) to \((2,-5)\), " Calculating, we get \(\sqrt{(2-7)^2 + (-5-4)^2} = \sqrt{25 + 81} = \sqrt{106}\). - Side 3: "\((2,-5)\) to \((0,2)\), " Calculating, we get \(\sqrt{(0-2)^2 + (2+5)^2} = \sqrt{4 + 49} = \sqrt{53}\).Since two sides are equal (both \(\sqrt{53}\)), triangle (a) is isosceles.

Step 3: Calculate Sides for Triangle (b)

For vertices (-1,-8), (0,-1), and (-4,-4), calculate the lengths of the sides:- Side 1: "\((-1,-8)\) to \((0,-1)\), " Calculating, we get \(\sqrt{(0+1)^2 + (-1+8)^2} = \sqrt{1 + 49} = \sqrt{50}\). - Side 2: "\((0,-1)\) to \((-4,-4)\), " Calculating, we get \(\sqrt{(-4-0)^2 + (-4+1)^2} = \sqrt{16 + 9} = \sqrt{25}\). - Side 3: "\((-4,-4)\) to \((-1,-8)\), " Calculating, we get \(\sqrt{(-1+4)^2 + (-8+4)^2} = \sqrt{9 + 16} = \sqrt{25}\).Since two sides are equal (both \(\sqrt{25}\)), triangle (b) is isosceles.

Step 4: Calculate Sides for Triangle (c)

For vertices (-7,4), (-3,10), and (1,3), calculate the lengths of the sides:- Side 1: "\((-7,4)\) to \((-3,10)\), " Calculating, we get \(\sqrt{(-3+7)^2 + (10-4)^2} = \sqrt{16 + 36} = \sqrt{52}\). - Side 2: "\((-3,10)\) to \((1,3)\), " Calculating, we get \(\sqrt{(1+3)^2 + (3-10)^2} = \sqrt{16 + 49} = \sqrt{65}\). - Side 3: "\((1,3)\) to \((-7,4)\), " Calculating, we get \(\sqrt{(-7-1)^2 + (4-3)^2} = \sqrt{64 + 1} = \sqrt{65}\).Since two sides are equal (both \(\sqrt{65}\)), triangle (c) is isosceles.

Key Concepts

Understanding Isosceles Triangles

An isosceles triangle is a type of triangle that has at least two equal sides. This property leads to two angles within the triangle also being equal. Why is this important? Because if we can identify these equal sides using mathematical tools, we confirm the triangle's classification as isosceles.

In coordinate geometry, we often have points (called vertices) that define the shape of a triangle. Using formulas, such as the distance formula, helps us determine if two sides are equal. When working with coordinate-based math problems, grasping the definition of an isosceles triangle is crucial, particularly when tasked with conjugating geometric principles to algebraic ones.

Here are some important points about isosceles triangles:

• They have at least two sides that are equal in length.
• The angles opposite those equal sides are also equal.
• Recognizing the shape helps in solving broader mathematical problems involving angles and side lengths.

The Role of Coordinate Geometry

Coordinate geometry, or analytic geometry, applies algebra to geometric problems. It involves using a coordinate system (like the Cartesian plane) to solve problems about shapes and figures. When given points, or vertices, this kind of geometry doesn't just help in drawing triangles but also in computing dimensions, lengths, and establishing relationships between shapes.

For a triangle problem, coordinate geometry allows us to:

• Locate and plot vertices on a graph.
• Use mathematical formulas, like the distance formula, to evaluate side lengths.
• Validate specific triangle types using numerical calculations.

Applying coordinate geometry lets students see how algebra and geometry intersect. This combination plays a significant role in effectively determining whether a triangle with given vertices is isosceles.

Identifying Triangle Vertices

In coordinate geometry, triangle vertices are essentially the points that define the triangle on a plane. Each vertex is represented by a coordinate pair such as \((x, y)\). Identifying these vertices is the first step when working through algebraic representation of geometric shapes.

To properly analyze a triangle using vertices:

• List each vertex with its corresponding coordinates. For example, vertices may be noted as A(0,2), B(7,4), C(2,-5).
• Understand how to visually plot these points to get a clear picture of the triangle’s shape.
• Utilize the distance formula to calculate side lengths by treating two vertices as points on the coordinate plane. This calculation gives insight into the nature of the triangle, such as determining if it is isosceles based on side length equality.

The procedure of analyzing triangle vertices serves as a foundational aspect of not just solving geometric problems but also reinforcing the interconnectedness of geometry and algebra.