Problem 5 - Calculus Early Transcendentals: Pearson New International Edition

Question

Show that the triangle whose vertices are \((5,3),(-2,4)\), and \((10,8)\) is isosceles.

Step-by-Step Solution

Answer

The triangle is isosceles because two sides are equal.

Step 1: Calculate Distance Between First Pair of Points

Using the distance formula, we calculate the distance between the points \((5,3)\) and \((-2,4)\). The distance formula is given by \[d = \sqrt{(x_2-x_1)^2 + (y_2-y_1)^2}\]Substituting in the coordinates, we get \[d = \sqrt{(-2-5)^2 + (4-3)^2} = \sqrt{(-7)^2 + (1)^2} = \sqrt{49 + 1} = \sqrt{50}\]

Step 2: Calculate Distance Between Second Pair of Points

Next, we compute the distance between the points \((-2,4)\) and \((10,8)\). Using the formula again, we have \[d = \sqrt{(10-(-2))^2 + (8-4)^2} = \sqrt{(10+2)^2 + (4)^2} = \sqrt{12^2 + 16} = \sqrt{144 + 16} = \sqrt{160}\]

Step 3: Calculate Distance Between Third Pair of Points

Lastly, we calculate the distance between the points \((5,3)\) and \((10,8)\). The distance is \[d = \sqrt{(10-5)^2 + (8-3)^2} = \sqrt{5^2 + 5^2} = \sqrt{25 + 25} = \sqrt{50}\]

Step 4: Compare the Distances

We find that two sides have the same length:\[\begin{align*}&\text{Distance between } (5,3) \text{ and } (10,8):\ \sqrt{50} \&\text{Distance between } (5,3) \text{ and } (-2,4):\ \sqrt{50} \ &\text{Distance between } (-2,4) \text{ and } (10,8):\ \sqrt{160}\end{align*}\]Since two sides are equal (\(\sqrt{50}\)), the triangle is isosceles.

Key Concepts

Distance Formula

The distance formula is an essential tool in geometry, especially when working with points in a coordinate system. It helps us find the distance between any two points in the plane. The formula is expressed as: \[d = \sqrt{(x_2-x_1)^2 + (y_2-y_1)^2}\] where \((x_1, y_1)\) and \((x_2, y_2)\) are the coordinates of the two points. By substituting these coordinates into the equation, we can calculate the distance between them.
This formula emanates from the Pythagorean Theorem applied to the difference of coordinates, considering it as the hypotenuse of a right triangle. Here's how you can use it effectively:

• First, identify the coordinates of two points you're interested in.
• Plug these into the distance formula.
• Solve under the square root and simplify to get your distance.

Understanding this will greatly aid in solving numerous problems in coordinate geometry.

Coordinate Geometry

Coordinate geometry, also known as analytic geometry, merges algebra and geometry to understand the positions of points on a plane. The most basic elements are the coordinates \((x, y)\) which specify the location of a point.
In the context of triangles, coordinate geometry allows us to calculate distances, midpoints, slopes, and understand the spatial relationships of points. Here's how it works:

• Dividing lines and angles: It uses algebraic principles to determine exact measurements and properties of geometric shapes.
• Calculating areas and perimeters: You can take known formulas and apply them using coordinates as variables.
• Finding specific types of triangles: With coordinate points, you can assess whether a triangle is equilateral, isosceles, or scalene by comparing distances.

Overall, coordinate geometry turns geometric exercises into manageable algebraic solutions.

Triangle Properties

Triangles, fundamental shapes in geometry, come with a variety of interesting properties. One is the classification by their sides:

• Scalene triangles: Have all sides of different lengths.
• Isosceles triangles: Have at least two equal sides.
• Equilateral triangles: Have all three sides of equal length.

When dealing with triangles on the coordinate plane, like in this exercise, identifying these properties requires the use of distance calculations.
Here’s how you determine if a triangle is isosceles:

• Find the lengths of each side using the distance formula.
• Compare these lengths to see if any two are equal.
• If at least two sides are the same, the triangle is isosceles.

This exercise illustrated that two sides of our triangle are equal, confirming its isosceles nature and showcasing the power of these fundamental properties.