# The set of vertices A ( 0 , 2 ) , B ( − 3 , − 1 ) and C ( − 4 , 3 ) form an isosceles triangle.

### Precalculus: Mathematics for Calcu...

6th Edition
Stewart + 5 others
Publisher: Cengage Learning
ISBN: 9780840068071

### Precalculus: Mathematics for Calcu...

6th Edition
Stewart + 5 others
Publisher: Cengage Learning
ISBN: 9780840068071

#### Solutions

Chapter 1.8, Problem 37E
To determine

## To show: The set of vertices A(0,2),B(−3,−1) and C(−4,3) form an isosceles triangle.

Expert Solution

### Explanation of Solution

Given information:

The set of vertices A(0,2),B(3,1) and C(4,3) .

Formula used:

Distance d(A,B) between two points A=(x1,y1) and B=(x2,y2) in the Cartesian plane is denoted by d(A,B)=(x2x1)2+(y2y1)2 .

Calculation:

Consider the provided set of vertices A(0,2),B(3,1) and C(4,3) .

It is known that in an isosceles triangle any two sides are equal.

Compute the length of sides of the triangle ABC that is AB,BC and CA .

Recall that the distance d(A,B) between two points A=(x1,y1) and B=(x2,y2) in the Cartesian plane is denoted by d(A,B)=(x2x1)2+(y2y1)2 .

Evaluate the distance between A(0,2) and B(3,1) .

d(A,B)=(30)2+(12)2=9+9=32

Next evaluate the distance between B(3,1) and C(4,3) .

d(B,C)=(4(3))2+(3(1))2=1+16=17

Evaluate the distance between A(0,2) and C(4,3) .

d(A,C)=(40)2+(32)2=16+1=17

Observe that d(B,C)=d(A,C) .

That is the distance between the point B(3,1) and the point C(4,3) is same as the distance the point A(0,2) and the point C(4,3) .

That is two sides of triangle are of equal length.

Therefore, triangle is an isosceles triangle.

Hence, it is shown that the set of vertices A(0,2),B(3,1) and C(4,3) form an isosceles triangle.

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