The area of triangle ABC and show that it is a right triangle using the converse of Pythagoras theorem.

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 40E
To determine

Expert Solution

Answer to Problem 40E

The area of triangle ABC is 20.5 sq. units .

Explanation of Solution

Given information:

The vertices of the triangle ABC A(6,7),B(11,3) and C(2,2) .

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

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 vertices of the triangle ABC A(6,7),B(11,3) and C(2,2) .

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)=(116)2+(3+7)2=(5)2+(4)2=25+16=41

Now, the length of BC will be calculated as,

d(B,C)=(112)2+(3+2)2=(9)2+(1)2=81+1=82

Length of AC will be calculated s,

d(A,C)=(62)2+(7+2)2=(4)2+(5)2=16+25=41

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

(d(A,B))2+(d(A,C))2=(d(B,C))2(41)2+(41)2=(82)241+41=8282=82

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. BC is equal to the sum of squares of other two sides, i.e. AB and AC, 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.

Now, BC is hypotenuse and AB and AC are base and height of the triangle ABC.

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(41)(41)=1241=20.5 sq. units

Thus, area of the triangle ABC is 20.5 sq. units .

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