Problem 17 - The Calculus with Analytic Geometry

Question

$$ \text { Show by means of slopes that the points }(-4,-1),\left(3, \frac{8}{3}\right),(8,-4) \text {, and }(2,-9) \text { are the vertices of a trapezoid. } $$

Step-by-Step Solution

Answer

The given points form a trapezoid because they have one pair of parallel sides.

Step 1 - Find the slope of each side

To determine if the points form a trapezoid, begin by finding the slopes of each side. Use the slope formula for two points \( m = \frac{y_2 - y_1}{x_2 - x_1} \)

Step 2 - Calculate Slope of Line 1

Calculate the slope between the points (−4,−1) and (3,\frac{8}{3}) as follows: \( m_1 = \frac{\frac{8}{3} - (-1)}{3 - (-4)} = \frac{8/3 + 3/3}{7} = \frac{11/3}{7} = \frac{11}{21}\)

Step 3 - Calculate Slope of Line 2

Calculate the slope between points (3,\frac{8}{3}) and (8,−4) as follows: \( m_2 = \frac{-4 - \frac{8}{3}}{8 - 3} = \frac{-4 - 8/3}{5} = \frac{(-12/3 - 8/3)}{5} = \frac{-20/3}{5} = -\frac{4}{3} \)

Step 4 - Calculate Slope of Line 3

Calculate the slope between points (8,−4) and (2,−9) as follows: \( m_3 = \frac{-9 - (-4)}{2 - 8} = \frac{-9 + 4}{-6} = \frac{-5}{-6} = \frac{5}{6} \)

Step 5 - Calculate Slope of Line 4

Calculate the slope between points (2,−9) and (−4,−1) as follows: \( m_4 = \frac{-1 - (-9)}{-4 - 2} = \frac{-1 + 9}{-6} = \frac{8}{-6} = -\frac{4}{3} \)

Step 6 - Determine Trapezoid

In a trapezoid, one pair of opposite sides must be parallel, i.e., have the same slope. In this case, the slopes of two sets of opposite sides are: \( m_2 = -\frac{4}{3} \) and \( m_4 = -\frac{4}{3} \) which means they are parallel.

Step 7: Verify Non-parallel Slopes

Ensure the other two sides are not parallel. Compare \( m_1 = \frac{11}{21} \) and \( m_3 = \frac{5}{6} \), these are not equal, therefore, these sides are not parallel.

Key Concepts

slope formula

To solve this exercise, we need to understand the slope formula. Simply put, the slope of a line measures its steepness. The mathematical formula for the slope between two points \((x_1, y_1)\) and \((x_2, y_2)\) is: \( m = \frac{y_2 - y_1}{x_2 - x_1} \).
Use this formula to find how one variable changes in relation to another.
In analytical geometry, this is important to determine relationships between points on the coordinate plane.
Practice applying this formula in different exercises to get a better grasp on its usage.

parallel lines

To identify a trapezoid using slopes, we need to look for parallel lines. In a trapezoid, only one pair of opposite sides is parallel.
If two lines are parallel, their slopes are equal. Therefore, when finding the slope of each side of a potential trapezoid, compare the slopes to see which lines are parallel.
Comparing slopes will help to confirm the parallel sides and thus determine if the figure is a trapezoid.
This concept is fundamental in geometry to classify figures accurately.

coordinate geometry

Coordinate geometry helps us analyze shapes using algebra. By placing figures on a coordinate plane, we can use mathematical formulas to investigate their properties.
In this exercise, we have vertices \((-4, -1), \left(3, \frac{8}{3}\right), (8, -4), (2, -9)\), and we will use their coordinates to determine the shape they form.
This approach simplifies problems and allows visualization of geometric shapes. Make sure to always plot the points correctly to avoid mistakes.

vertices identification

To start solving the exercise, identify and label the vertices of the shape given in the problem.
In any geometric problem involving shapes, knowing the exact positions of the vertices is a must.
Begin by labeling each point \((-4, -1) = A, \left(3, \frac{8}{3}\right) = B, (8, -4) = C, (2, -9) = D\).
Once labeled, use these to find the slopes between each pair of vertices systematically.

analytical geometry

Analytical geometry, also known as coordinate geometry, combines algebra and geometry to solve problems involving geometric figures.
It utilizes coordinates to study shapes, making precise calculations possible.
By applying slope calculations and comparing them, we can analyze and classify the shape formed by the points.
Familiarize yourself with conversion between graphical representations and algebraic methods to use analytical geometry effectively.