Problem 11 - Algebra and Trigonometry with Analytic Geometry

Question

Exer. 7-10: Use slopes to show that the points are vertices of the specified polygon. If three consecutive vertices of a parallelogram are \(A(-1,-3), B(4,2)\), and \(C(-7,5)\), find the fourth vertex.

Step-by-Step Solution

Answer

The fourth vertex is \((-12, 0)\).

Step 1: Calculate Slope of AB

Find the slope of line segment AB using the formula for the slope between two points, \(m = \frac{y_2 - y_1}{x_2 - x_1}\) where \((x_1, y_1) = A(-1, -3)\) and \((x_2, y_2) = B(4, 2)\).\[m_{AB} = \frac{2 - (-3)}{4 - (-1)} = \frac{5}{5} = 1\]

Step 2: Calculate Slope of BC

Find the slope of line segment BC using the same slope formula with points \((x_1, y_1) = B(4, 2)\) and \((x_2, y_2) = C(-7, 5)\).\[m_{BC} = \frac{5 - 2}{-7 - 4} = \frac{3}{-11}\]

Step 3: Determine the Opposite Slope for DC

In a parallelogram, opposite sides are parallel. Thus, the slope of segment DC must match the slope of AB. So, \(m_{DC} = 1\). Assume point D is \((x, y)\) and calculate \(y\) in terms of \(x\).

Step 4: Use Slope of CD to Find Coordinates

Use the slope condition \(m_{DC} = 1\) to set up the equation using point C \((-7, 5)\) and unknown D \((x, y)\).\[1 = \frac{y - 5}{x + 7}\]Solve for \(y\):\[y = x + 7 + 5 = x + 12\]

Step 5: Calculate Slope of AD

Since AD is parallel to BC, AD should also have the same slope as BC. Use the condition \(m_{AD} = \frac{3}{-11}\). Using point A \((-1, -3)\) and assumed point D \((x, y)\),\[\frac{y - (-3)}{x - (-1)} = \frac{3}{-11}\]Substitute \(y = x + 12\) into the equation:\[\frac{(x + 12) + 3}{x + 1} = \frac{3}{-11}\]

Step 6: Solve for x

Resolve the equation from Step 5:\[\frac{x + 15}{x + 1} = \frac{3}{-11}\]Cross-multiply to clear the fraction:\[-11(x + 15) = 3(x + 1)\]Simplify and solve for \(x\):\[-11x - 165 = 3x + 3\]Combine terms:\[-14x = 168\]Divide by \(-14\):\[x = -12\]

Step 7: Calculate y from x

Substitute \(x = -12\) back into the equation \(y = x + 12\) from Step 4.\[y = -12 + 12 = 0\]

Step 8: Verify the Quadrilateral is a Parallelogram

Check if the fourth point \(D(-12, 0)\) creates parallel opposite sides with the given points using their slopes, verifying it completes a parallelogram.

Key Concepts

Parallelogram

A parallelogram is a type of quadrilateral, specifically one where opposite sides are both equal in length and parallel. This means two sides of the parallelogram have the same slope, confirming their parallel nature.

• Opposite sides are equal in length.
• Opposite sides are parallel (same slope).
• Consecutive angles in a parallelogram are supplementary (sum to 180°).

Knowing these properties helps identify or complete a parallelogram when given certain vertices. In this exercise, calculating slopes allowed us to use these properties to find the fourth vertex.

Slope

The slope of a line in geometry represents its steepness, often described as 'rise over run'. It is calculated using two points from the line:\[ m = \frac{y_2 - y_1}{x_2 - x_1} \]where \( (x_1, y_1) \) and \( (x_2, y_2) \) are any two points on the line. By determining the slopes of the lines forming a parallelogram, we can check:

• If two lines are parallel by comparing their slopes.
• The particular direction and steepness of the lines.

The importance of the slope here was to ascertain parallelism, crucial for confirming the structure is indeed a parallelogram.

Quadrilateral

Quadrilaterals are polygons with four edges and vertices. They can take various forms, such as squares, rectangles, and parallelograms. Key aspects include:

• Quadrilaterals have four sides and angles.
• The sum of internal angles in any quadrilateral is 360°.
• Special types include parallelograms, rectangles, and squares based on their side lengths and angles.

Parallelograms are specific quadrilaterals where opposite sides are parallel. Understanding these properties helps in identifying and verifying quadrilaterals and their subtypes geometrically.

Coordinate Geometry

Coordinate geometry, or analytic geometry, is the study of geometry using a coordinate system. By placing figures into a coordinate plane, we can use algebraic equations to solve geometric problems. In this context:

• Points are represented by coordinates: \( (x, y) \).
• Equations can determine distances, slopes, and shapes.
• Helps in visualizing geometrical problems accurately and efficiently.

This method allows for an algebraic approach to classic geometry. In the task, coordinate geometry was applied to find the missing vertex of a parallelogram by using slope equations and the Cartesian plane properties.