Advanced Mathematical Concepts: Precalculus with Applications, Student Edition
Advanced Mathematical Concepts: Precalculus with Applications, Student Edition
1st Edition
ISBN: 9780078682278
Author: McGraw-Hill, Berchie Holliday
Publisher: Glencoe/McGraw-Hill
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Chapter 10, Problem 13SGA
To determine

To find: whether the points A(−5,−2),B(3,4),C(10,3),D(2,−3) are the vertices of a parallelogram.

Expert Solution & Answer
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Answer to Problem 13SGA

The given points are the vertices of a parallelogram.

Explanation of Solution

Given information:

The points A(−5,−2),B(3,4),C(10,3),D(2,−3) .

To show the points A(−5,−2),B(3,4),C(10,3),D(2,−3) are the vertices of a parallelogram, it is necessary to show that the opposite sides of a parallelogram should be parallel and congruent.

Thus,

  Advanced Mathematical Concepts: Precalculus with Applications, Student Edition, Chapter 10, Problem 13SGA

Since, the sides DA,DB are parallel.

Now,

To determine DA∥CB , it is require to find slope.

Slope of side DA can be calculated by the formula,

  m=y2−y1x2−x1 .

Here,

  y2=3,y1=4,x1=3,x2=10 .

Substitute the value in the given formula,

  m=y2−y1x2−x1=3−410−3=−17

Similarly the Slope of side CB is,

  m=y2−y1x2−x1=−3−(−2)2−(−5)=−17

Since the slopes are equal. Therefore, DA∥CB .

To determine the opposite sides DA,CB and AB,CD of parallelogram are equal we use distance formula,

  DA=(x2−x1)2+(y2−y1)2 .

Substitute the values in the given formula,

  CB=(x2−x1)2+(y2−y1)2=(2−(−5))2+(−3−(−2))2=72+(−1)2=50

The measures of opposite sides DA,CB are equal. Therefore, DA∥CB .

Now,

  AB=(x2−x1)2+(y2−y1)2=(10−2)2+(3−(−3))2=82+(6)2=100

  CD=(x2−x1)2+(y2−y1)2=(3−(−5))2+(4−(−2))2=82+(6)2=100

The measures of sides AB,CD are equal. Therefore, AB∥CD .

Now, the length of the diagonal is,

  AC=(x2−x1)2+(y2−y1)2=(−5−10)2+(−2−3)2=(−15)2+(−5)2=250

Similarly, the length of BD is,

  BD=(x2−x1)2+(y2−y1)2=(3−2)2+(4−(−3))2=(1)2+(7)2=50

Since, the diagonal AC≠BD .

Therefore, the given points are the vertices of a parallelogram.

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Chapter 10 Solutions

Advanced Mathematical Concepts: Precalculus with Applications, Student Edition

Ch. 10.1 - Prob. 1CFUCh. 10.1 - Prob. 2CFUCh. 10.1 - Prob. 3CFUCh. 10.1 - Prob. 4CFUCh. 10.1 - Prob. 5CFUCh. 10.1 - Prob. 6CFUCh. 10.1 - Prob. 7CFUCh. 10.1 - Prob. 8CFUCh. 10.1 - Prob. 9CFUCh. 10.1 - Prob. 10CFU
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