BuyFind

Precalculus: Mathematics for Calcu...

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

Precalculus: Mathematics for Calcu...

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

Solutions

Chapter 1.8, Problem 39E

(a)

To determine

To verify: The triangleABC is a right triangle using converse of Pythagoras theorem.

Expert Solution

Explanation of Solution

Given information:

The figure,

  Precalculus: Mathematics for Calculus - 6th Edition, Chapter 1.8, Problem 39E , additional homework tip  1

Formula used:

Converse of Pythagoras theorem states that if the sum of squares of two sides of a triangle is equal to the square of longest side of the triangle, then the triangle is a right triangle.

Distance formula between two points X(a1,b1) and Y(a2,b2) is calculated as,

  d(XY)=(a2a1)2+(b2b1)2

Proof:

Consider the given figure,

  Precalculus: Mathematics for Calculus - 6th Edition, Chapter 1.8, Problem 39E , additional homework tip  2

In the above figure, the coordinates of the vertices of the triangle ABC are A(2,2),B(3,1) and C(3,3).

Recall that the distance formula between two points X(a1,b1)and Y(a2,b2)is calculated as,

  d(XY)=(a2a1)2+(b2b1)2

So, length of AB will be calculated as.

  d(A,B)=(32)2+(12)2=(1)2+(3)2=1+9=10

Now, the length of BC will be calculated as,

  d(B,C)=(33)2+(3+1)2=(6)2+(2)2=36+4=40

Length of AC will be calculated s,

  d(A,C)=(32)2+(32)2=(5)2+(5)2=25+25=50

Now, calculate the sum of squares of AB and BC as,

  (d(A,B))2+(d(B,C))2=(d(A,C))2(10)2+(40)2=(50)210+40=5050=50

Recall the converse of Pythagoras theorem if the sum of squares of two sides of a triangle is equal to the square of longest side of the triangle, then the triangle is a right triangle.

Here, square of longest side i.e. AC is equal to the sum of squares of other two sides, i.e. AB and BC, so, the given triangle ABC is a right triangle.

Thus, using converse of Pythagoras theorem it is proved that the triangle ABC is a right triangle.

(b)

To determine

To calculate: The area of triangle ABC.

Expert Solution

Answer to Problem 39E

The area of triangle ABC is 10 sq. units.

Explanation of Solution

Given information:

The figure,

  Precalculus: Mathematics for Calculus - 6th Edition, Chapter 1.8, Problem 39E , additional homework tip  3

Formula used:

Area of a triangle is half into the product of its base and height. So, if b is the base and h is the height of the triangle, then area of triangle is expressed as,

  Area=12bh

Calculation:

Consider the given figure,

  Precalculus: Mathematics for Calculus - 6th Edition, Chapter 1.8, Problem 39E , additional homework tip  4

In (a) part, lengths of sides of triangle are calculated as,

Length of AB =10 units.

Length of BC =40 units.

Length of AC =50 units.

Since, AC is longest, so, it is hypotenuse and AB is shortest, so, AB is the perpendicular (height) and BC is base of the triangle.

Recall area of a triangle is half into the product of its base and height. So, if b is the base and h is the height of the triangle, then area of triangle is expressed as,

  Area=12bh

Apply it,

  Area=12bh=12(40)(10)=12(400)

Simplify it further as,

  Area=12(400)=12(20)=10 sq. units

Thus, area of the triangle ABC is 10 sq. units.

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