△ ABC, c^2 > a^2+b^2 where c is the length of the longest side
1.
Given
2.
△ XYZ a right triangle so that x is the length of the longest side and the measures of the legs are a and b
2.
Construct
3.
a^2+b^2 = x^2
3.
Pythagorean Theorem
4.
c^2 > x^2
4.
Substitution Property
5.
c > x
5.
A property of square roots
6.
m∠ C > m∠ X
6.
Converse of the Hinge Theorem
7.
m∠ X = 90^(∘)
7.
Definition of right angle
8.
m∠ C > 90^(∘)
8.
Substitution Property
9.
∠ C is an obtuse angle
9.
Definition of an obtuse angle
10.
△ ABC is an obtuse triangle
10.
Definition of an obtuse triangle
Practice makes perfect
We are asked to write a two-column proof of the following theorem.
Theorem 8.6
If the square of the length of the longest side of a triangle is less than the sum of the squares of the lengths of the other two sides, then the triangle is an acute triangle.
Let's begin by considering △ ABC shown below.
We can now rewrite the given information and the statement what we want to prove.
Given: & △ ABC, c^2 > a^2+b^2 wherec is the
& length of the longest side
Prove: & △ ABC is obtuse
Additionally, let's consider a right triangle XYZ so that its legs have measures equal to a and b.
By the Pythagorean Theorem, we have that a^2 + b^2 = x^2. Since we also know that c^2 < a^2+ b^2, this leads us to the following inequality.
lc^2 > a^2+ b^2 a^2 + b^2 = x^2 ⇒ c^2 > x^2
We can take a square root of both sides of the inequality.
Let's now include all the relations between △ ABC and △ XYZ in the graph.
Side opposite ∠ C is longer than side opposite ∠ X.
By Converse of the Hinge Theorem, we conclude that m∠ C > m∠ X. Since we also know that m∠ X=90^(∘), we have that ∠ C is an obtuse angle.
m∠ C > 90 ^(∘)
Triangle ABC has an obtuse angle, so it is an obtuse triangle.
Two-Column Proof
In the following table, we will summarize the proof we did above.
Statements
Reasons
1.
△ ABC, c^2 > a^2+b^2 where c is the length of the longest side
1.
Given
2.
△ XYZ a right triangle so that x is the length of the longest side and the measures of the legs are a and b
