Problem 14 - Intermediate Algebra

Question

Verify that the points \((0,3),(2,-3)\), and \((-4,-5)\) are vertices of an isosceles triangle.

Step-by-Step Solution

Answer

Yes, the points form an isosceles triangle with two equal sides of length \(\sqrt{40}\).

Step 1: Calculate Distance Between Points (0,3) and (2,-3)

To verify if the given points are vertices of an isosceles triangle, we need to check the distances between each pair of points. Use the distance formula: \[d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}\]For points \((0,3)\) and \((2,-3)\), substitute: \[d = \sqrt{(2 - 0)^2 + (-3 - 3)^2} = \sqrt{2^2 + (-6)^2} = \sqrt{4 + 36} = \sqrt{40}\].

Step 2: Calculate Distance Between Points (2,-3) and (-4,-5)

Now calculate the distance between the next pair of points (2,-3) and (-4,-5) using the same distance formula:\[d = \sqrt{(-4 - 2)^2 + (-5 + 3)^2} = \sqrt{(-6)^2 + (-2)^2} = \sqrt{36 + 4} = \sqrt{40}\].

Step 3: Calculate Distance Between Points (0,3) and (-4,-5)

Finally, calculate the distance between the remaining pair of points (0,3) and (-4,-5):\[d = \sqrt{(-4 - 0)^2 + (-5 - 3)^2} = \sqrt{(-4)^2 + (-8)^2} = \sqrt{16 + 64} = \sqrt{80}\].

Step 4: Compare Distances to Confirm Isosceles Triangle

For a triangle to be isosceles, at least two sides must be equal in length. We have calculated the following side lengths: * \(\sqrt{40}\) (from steps 1 and 2)* \(\sqrt{80}\) (from step 3).Since \(\sqrt{40} = \sqrt{40}\), two sides of the triangle are equal, confirming that it is an isosceles triangle.

Key Concepts

Distance Formula

Understanding the distance formula is crucial when dealing with coordinate geometry, especially for tasks like verifying the type of triangle formed by given vertices. We use the distance formula to calculate the distance between two points in a plane. This formula is derived from the Pythagorean theorem. It can be written as: \[d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}\]. This formula helps determine how far apart two points \(x_1, y_1\) and \(x_2, y_2\) on a plane are. By substituting in the coordinates of the points, we can compute the exact distance between them.

In our exercise, this method was used to find out if the triangle formed by the points \(0,3\), \(2,-3\), and \(-4,-5\) is isosceles, by comparing distances between these points.

Coordinate Geometry

Coordinate geometry, or analytic geometry, provides a bridge between algebra and geometry through graphs and equations. This powerful branch of mathematics allows us to use algebraic procedures to address geometric problems such as determining distance, angle, and area.

It involves points being represented by sets of numbers, or coordinates, which gives us flexibility in calculations and problem-solving. Hence, tasks like checking the properties of triangles on a coordinate plane become more straightforward. In this exercise, coordinate geometry enables us to straightforwardly calculate side lengths of the triangle in question using algebraic methods, rather than traditional geometric constructions. This makes verifying whether the triangle is isosceles much simpler, by directly applying calculations and comparisons.

Isosceles Triangle Properties

An isosceles triangle is defined by having at least two sides of equal length. This property leads to other distinctive characteristics. For example, the angles opposite the equal sides are also equal. These properties can be exploited to solve many geometric problems.

When verifying a triangle's isosceles nature using coordinates, it boils down to ensuring two sides have the same length. This length can be found using the distance formula. In our example, by comparing calculated distances between \(0,3\) to \(2,-3\), \(2,-3\) to \(-4,-5\), and \(0,3\) to \(-4,-5\), we identified that two sides have lengths of \(\sqrt{40}\). This identical length confirms that the triangle formed by the given points is isosceles.