Problem 104 - A Graphical Approach to College Algebra

Question

Solve each problem. Triangles can be classified by their sides. (a) An isosceles triangle has at least two sides of equal length. Determine whether the triangle with vertices (0,0),(3,4), and (7,1) is isosceles. (b) An equilateral triangle has all sides of equal length. Determine whether the triangle with vertices (-1,-1), (2,3) , and (-4,3) is equilateral. (c) Determine whether a triangle having vertices (-1,0) (1,0) and (0, \sqrt{3}) is isosceles, equilateral, or neither. (d) Determine whether a triangle having vertices (-3,3) (-2,5) and (-1,3) is isosceles, equilateral, or neither.

Step-by-Step Solution

Answer

(a) Isosceles and equilateral; (b) Isosceles; (c) Equilateral; (d) Isosceles.

Step 1: Calculate distance between points for (a)

To check if the triangle with vertices \((0,0)\), \((3,4)\), and \((7,1)\) is isosceles, we calculate the distances between each pair of points. Use the distance formula \(d = \sqrt{(x_2-x_1)^2 + (y_2-y_1)^2}\).1. Distance between \((0,0)\) and \((3,4)\):\[\sqrt{(3-0)^2 + (4-0)^2} = \sqrt{9 + 16} = 5\]2. Distance between \((3,4)\) and \((7,1)\):\[\sqrt{(7-3)^2 + (1-4)^2} = \sqrt{16 + 9} = 5\]3. Distance between \((7,1)\) and \((0,0)\):\[\sqrt{(7-0)^2 + (1-0)^2} = \sqrt{49 + 1} = 5\]All three distances (5, 5, and 5) are equal, indicating an equilateral triangle or by definition, isosceles.

Step 2: Conclusion for (a)

Since all sides are equal, the triangle is not only isosceles but also equilateral. According to the given categorization, it qualifies as isosceles.

Step 3: Calculate distance between points for (b)

To determine if a triangle with vertices \((-1,-1)\), \((2,3)\), and \((-4,3)\) is equilateral, calculate the side lengths:1. Distance between \((-1,-1)\) and \((2,3)\):\[\sqrt{(2+1)^2 + (3+1)^2} = \sqrt{3^2 + 4^2} = \sqrt{9 + 16} = 5\]2. Distance between \((2,3)\) and \((-4,3)\):\[\sqrt{(-4-2)^2 + (3-3)^2} = \sqrt{(-6)^2 + 0^2} = \sqrt{36} = 6\]3. Distance between \((-4,3)\) and \((-1,-1)\):\[\sqrt{(-1+4)^2 + (-1-3)^2} = \sqrt{3^2 + (-4)^2} = \sqrt{9 + 16} = 5\]

Step 4: Conclusion for (b)

Since not all side lengths are equal \((5, 6, 5)\), the triangle is not equilateral. However, two sides are equal, so it is isosceles.

Step 5: Calculate distance between points for (c)

For the triangle with vertices \((-1,0)\), \((1,0)\), and \((0, \sqrt{3})\), calculate the side lengths:1. Distance between \((-1,0)\) and \((1,0)\):\[\sqrt{(1+1)^2 + (0-0)^2} = \sqrt{4} = 2\]2. Distance between \((1,0)\) and \((0, \sqrt{3})\):\[\sqrt{(0-1)^2 + (\sqrt{3}-0)^2} = \sqrt{1 + 3} = \sqrt{4} = 2\]3. Distance between \((-1,0)\) and \((0, \sqrt{3})\):\[\sqrt{(0+1)^2 + (\sqrt{3}-0)^2} = \sqrt{1 + 3} = \sqrt{4} = 2\]

Step 6: Conclusion for (c)

With all sides equal, the triangle is both isosceles and equilateral.

Step 7: Calculate distance between points for (d)

For the triangle with vertices \((-3,3)\), \((-2,5)\), and \((-1,3)\), calculate the side lengths:1. Distance between \((-3,3)\) and \((-2,5)\):\[\sqrt{(-2+3)^2 + (5-3)^2} = \sqrt{1^2 + 2^2} = \sqrt{5}\]2. Distance between \((-2,5)\) and \((-1,3)\):\[\sqrt{(-1+2)^2 + (3-5)^2} = \sqrt{1 + 4} = \sqrt{5}\]3. Distance between \((-1,3)\) and \((-3,3)\):\[\sqrt{(-3+1)^2 + (3-3)^2} = \sqrt{(-2)^2} = 2\]

Step 8: Conclusion for (d)

Since two sides are \(\sqrt{5}\) and one is 2, this triangle is isosceles because two sides are equal.

Key Concepts

Isosceles Triangle

An isosceles triangle is a special type of triangle with at least two equal sides. This property gives the isosceles triangle its distinct shape and symmetry, also making it easier to calculate angles and other properties.

To identify an isosceles triangle, you need to calculate the side lengths using the distance formula, which helps compare them easily. Once you find any two sides that are equal, you can confirm that the triangle is isosceles.

Isosceles triangles have some unique properties:

• Two sides of equal length
• Two angles opposite the equal sides which are also equal
• Can be found and confirmed using the Pythagorean theorem when sides are known

In exercises involving coordinates or given side lengths, identifying an isosceles triangle often involves proving that two distances among the vertices are equal.

Equilateral Triangle

An equilateral triangle is a triangle in which all three sides are identical in length. Because all sides are equal, all its angles are also equal, each being precisely 60 degrees.

To verify if a triangle is equilateral using coordinate geometry, you calculate the distances between each pair of vertices. If all three calculated distances are equal, then the triangle is equilateral. This ensures a perfectly symmetrical shape every time.

Key properties of equilateral triangles include:

• Three sides of equal length
• Three internal angles equal at 60 degrees each
• Regular symmetric properties, which simplifies many calculations

Equilateral triangles are a special subset of isosceles triangles since they share the property of having at least two equal sides.

Distance Formula

The distance formula is a foundational element of geometry especially when dealing with coordinates on a plane. It is used to calculate the distance between two points with known coordinates.

The formula is given by: \[ d = \sqrt{(x_2-x_1)^2 + (y_2-y_1)^2} \] where the points are \(x_1, y_1\) and \(x_2, y_2\). This calculation is vital for determining the lengths of the sides of a triangle, which is crucial for classifying the type of triangle.

Benefits of understanding and applying the distance formula include:

• Accurate determination of lengths in a coordinate plane
• Essential tool for geometry and trigonometry calculations
• Helps in classifying triangles and understanding their properties

Learning to apply this formula effectively allows for precise and easy classification of triangles into isosceles, equilateral, or neither.