/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 94 Use your knowledge of the slopes... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 3: Problem 94

Use your knowledge of the slopes of parallel and perpendicular lines. Is the figure with vertices at \((-11,-5),(-2,-19),(12,-10),\) and \((3,4)\) a parallelogram? Is it a rectangle? (Hint: A rectangle is a parallelogram with a right angle.)

Short Answer

Expert verified
The figure is a rectangle because it is a parallelogram with perpendicular adjacent sides.

Step by step solution

01

Find the Slopes of Opposite Sides

First, calculate the slopes of the sides formed by the vertices. Use the slope formula \(\text{slope} = \frac{y_2 - y_1}{x_2 - x_1}\). Calculate the slopes for each pair of points: \((-11, -5)\) to \((-2, -19)\), \((-2, -19)\) to \(12, -10)\), \(12, -10)\) to \(3, 4)\), and \(3, 4)\) to \((-11, -5)\).
02

Calculate Slope for Side 1

For points \((-11, -5)\) and \((-2, -19)\), the slope is: \[ \text{slope} = \frac{-19 - (-5)}{-2 - (-11)} = \frac{-14}{9} = -\frac{14}{9} \]
03

Calculate Slope for Side 2

For points \((-2, -19)\) and \(12, -10)\), the slope is: \[ \text{slope} = \frac{-10 - (-19)}{12 - (-2)} = \frac{9}{14} \]
04

Calculate Slope for Side 3

For points \(12, -10)\) and \(3, 4)\), the slope is: \[ \text{slope} = \frac{4 - (-10)}{3 - 12} = \frac{14}{-9} = -\frac{14}{9} \]
05

Calculate Slope for Side 4

For points \(3, 4)\) and \((-11, -5)\), the slope is: \[ \text{slope} = \frac{-5 - 4}{-11 - 3} = \frac{-9}{-14} = \frac{9}{14} \]
06

Compare Slopes for Opposite Sides

Compare the slopes of the opposite sides. Sides \((-11, -5)\) to \((-2, -19)\) and \(12, -10)\) to \(3, 4)\) both have a slope of \(-\frac{14}{9}\). Similarly, sides \((-2, -19)\) to \(12, -10)\) and \(3, 4)\) to \((-11, -5)\) both have a slope of \(\frac{9}{14}\). This means they are parallel.
07

Determine if It Is a Parallelogram

Since opposite sides are parallel, the figure is a parallelogram.
08

Check Perpendicularity of Adjacent Sides

To check if the parallelogram is a rectangle, multiply the slopes of two adjacent sides. If the product is \(-1\), the sides are perpendicular. Calculate \(-\frac{14}{9} \times \frac{9}{14} = -1\).
09

Conclusion

Since the product of the slopes of adjacent sides is \(-1\), the adjacent sides are perpendicular, indicating the figure is a rectangle.

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Slope Formula
The slope of a line represents its steepness and direction. The formula to calculate the slope between two points on a plane, \((x_1, y_1)\) and \((x_2, y_2)\), is given by:
\[ \text{slope} = \frac{y_2 - y_1}{x_2 - x_1} \]
This formula helps compare the inclines of different lines:
  • If two lines are parallel, their slopes are equal.
  • If two lines are perpendicular, the product of their slopes is \(-1\).
In the exercise, you calculated the slopes of the sides of a given quadrilateral and identified it as a parallelogram by comparing the slopes of opposite sides. By confirming the condition for perpendicular lines, you also concluded that the figure is a rectangle.
Parallelogram
A parallelogram is a four-sided figure where opposite sides are parallel. This means:
  • The slopes of opposite sides are equal.
  • Opposite angles are equal.
  • The diagonals bisect each other.
To determine if a figure is a parallelogram using coordinate geometry, you can calculate the slopes of all four sides. If each pair of opposite sides has equal slopes, the figure is a parallelogram. In this exercise, by using the slope formula, you found that the opposite sides have equal slopes, \(-\frac{14}{9}\) and \(\frac{9}{14}\), which confirmed the figure is a parallelogram.
Rectangle Geometry
A rectangle is a type of parallelogram with right angles at all four corners. This fact gives it these unique properties:
  • All angles are 90 degrees.
  • Opposite sides are equal and parallel.
  • The diagonals are equal in length and bisect each other.
To verify if a parallelogram is a rectangle in coordinate geometry, check if adjacent sides are perpendicular. You do this by multiplying their slopes. If the result is \-1\, the sides are perpendicular. In the problem, you multiplied the slopes of adjacent sides, \(-\frac{14}{9}\) \times \(\frac{9}{14}\), and the product was \-1\. Thus, the figure is confirmed as a rectangle.
Coordinate Geometry
Coordinate geometry combines algebra and geometry using a coordinate system to solve geometric problems. Important concepts include:
  • Using coordinates \((x, y)\) to denote points on a plane.
  • Finding the distance between points with the distance formula, \[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]
  • Calculating the midpoint of a segment with \[ \text{midpoint} = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right) \]
  • Determining slopes and analyzing the orientation of lines.
In the given problem, you used coordinate geometry techniques: calculating slopes to establish the nature of a figure, confirming it as a parallelogram and validating it as a rectangle by checking the perpendicularity of sides.

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

Solve each problem. Forensic scientists use the lengths of certain bones to calculate the height of a person. Two bones often used are the tibia \((t),\) the bone from the ankle to the knee, and the femur \((r),\) the bone from the knee to the hip socket. A person's height ( \(h\) ) in centimeters is determined from the lengths of these bones by using functions defined by the following formulas. For men: \(h(r)=69.09+2.24 r\) or \(h(t)=81.69+2.39 t\) For women: \(h(r)=61.41+2.32 r\) or \(h(t)=72.57+2.53 t\) A.Find the height of a man with a femur measuring \(56 \mathrm{cm} .\) B. Find the height of a man with a tibia measuring \(40 \mathrm{cm} .\) C. Find the height of a woman with a femur measuring \(50 \mathrm{cm} .\) D. Find the height of a woman with a tibia measuring \(36 \mathrm{cm} .\) (PICTURE CANT COPY)

An equation that defines \(y\) as a function fof \(x\) is given. (a) Solve for \(y\) in terms of \(x,\) and \(r e-\) place \(y\) with the function notation \(f(x) .\) (b) Find \(f(3) .\) See Example 6. Fill in each blank with the correct response. The equation \(2 x+y=4\) has a straight _____ as its graph. One point that lies on the graph is \((3\),____ ).If we solve the equation for \(y\) and use function notation, we obtain \(f(x)=\) _____ . For this function, \(f(3)=\) _____ ,meaning that the point ( ____,_____ ) lies on the graph of the function.

An equation that defines \(y\) as a function fof \(x\) is given. (a) Solve for \(y\) in terms of \(x,\) and \(r e-\) place \(y\) with the function notation \(f(x) .\) (b) Find \(f(3) .\) See Example 6. $$ x-4 y=8 $$

Diluting Solutions. How much water should be added to 20 ounces of a \(15 \%\) solution of alcohol to dilute it to a \(10 \%\) alcohol solution?

For each situation, (a) write an equation in the form \(y=m x+b,(b)\) find and interpret the ordered pair associated with the equation for \(x=5,\) and \((c)\) answer the question. A rental car costs \(\$ 50\) plus \(\$ 0.20\) per mile. Let \(x\) represent the number of miles driven and \(y\) represent the total charge to the renter. How many miles was the car driven if the renter paid \(\$ 84.60 ?\)
See all solutions

Recommended explanations on Math Textbooks

Discrete Mathematics

Read Explanation

Decision Maths

Read Explanation

Probability and Statistics

Read Explanation

Logic and Functions

Read Explanation

Geometry

Read Explanation

Applied Mathematics

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.