This lesson explains the concepts of Pythagorean Theorem Converse
and how to prove converse of
the Pythagorean Theorem geometrically. It is such designed to enable you to do home work and
to solve problems involving right triangles. You may recall from earlier learning that the
formula for the Pythagorean Theorem is
a^{2} + b^{2} = c^{2} and it can also be used to check whether
a given triangle is an acute angle triangle, a right angle triangle or an obtuse angle triangle.
The presentation covering such content will be done by the instructor in own handwriting, using
video and with the help of several examples with solution. This will help you understand
important geometric relationships to solve problems from day to day life situations based
on the above concepts, and how to make use of them.
People who saw this lesson also found the following lessons useful:
(Continued from above)The Converse
of the Pythagorean Theorem
For any three positive numbers a, b, and c such that
a^{2} + b^{2} = c^{2}, there exists a triangle with sides a,
b and c, and every such triangle has a right angle between the sides of lengths a and b.
Theorem
If the square of the length of one side of a triangle is equal to the sum of the squares of the lengths of the other two sides, then the triangle is a right triangle.
Given ABC with
AB = c, BC = a, CA = b, and c^{2} =
a^{2} + b^{2}
Prove ABC is a right triangle with C the right angle.
Proof Draw DEF with
EF = a, FD = b, and F a right angle.
=> DE^{2} = a^{2} + b^{2}, DE^{2} = c^{2} and DE
= c
=>
ABCDEF
(by SSS)
=>
CF
=>
C
is a right angle.
=>
ABC
is a right triangle.
Another Theorem
A triangle is a right triangle if and only if the square of the length of the longest side is equal to the sum of the squares of the lengths of the other two sides.
A corollary of the Pythagorean Theorem’s converse is a simple means of determining whether a triangle is right, obtuse, or acute, where c is chosen to be the longest of the three sides.
The Converse of the Pythagorean Theorem: as seen above, and the Pythagorean Theorem can be used to check whether a given triangle is an acute angle triangle, a right angle triangle or an obtuse angle triangle.
For a triangle with sides a, b and c and c is the longest side then:
• If c^{2} < a^{2} + b^{2}, it is an acuteangle triangle, i.e. the angle facing side c is an acute angle.
• If c^{2} = a^{2} + b^{2}, it is a rightangle triangle, i.e. the angle facing side c is a right angle.
• If c^{2} > a^{2} + b^{2}, it is an obtuseangle triangle, i.e. the angle facing side c is an obtuse angle.
Further on you’ll look into some illustrative examples with solution:
Example 1: Given find out whether a triangle with sides 4 cm, 6 cm and 8 cm is an acuteangled, rightangled or obtuseangled triangle.
Step 1: First choose the two shorter sides, say a and b and the longest side to be c.
Step 2: a = 4, b = 6 and c = 8 (given)
Step 3: a^{2} + b^{2} = 4^{2} + 6^{2} = 16 + 36 = 52, and c^{2} = 8^{2} = 64
Step 4: 64 > 52, the pattern being > c^{2} > a^{2} + b^{2}
Therefore, it is an obtuseangle triangle.
The triangle is an obtuseangle triangle, as the final answer.
Example 2: Given can a triangle have more than one obtuse angle?
You know that an obtuse triangle has one obtuse angle i.e. it will be exceeding 90
and the longest side is opposite to the obtuse angle. In the obtuse triangle shown below,
is the obtuse angle.
Step 1: Say the angles of the triangle ABC will be a, b
and c. Angle a i.e.
be the obtuse angle.
Step 2: The sum of three angles in a triangle is 180
i.e. + b + c = 180.
Step 3: Since
is > 90,
then b + c must be less than 90.When the sum of angle b and c, is less that 90, obviously each one i.e. b and c shall be less than 90
Therefore, b and c must be acute angles.
Conclusion: No, a triangle can have only one obtuse angle, as the final answer.
Example 3: Given find out whether a triangle with side lengths of 10 cm, 12 cm and 15 cm is an acuteangle, rightangle or obtuseangle triangle.
Say the triangle is with sides a, b and c.
Out of this
c is the longest side.
Step 1: In the given dimensions, assume a = 10, b = 12 and c
= 15 cm
Step 2: a^{2} + b^{2} = 10^{2} + 12^{2} = 100 + 144 = 244 Step 3: Given, c = 15 i.e. c^{2} = 15^{2} = 225
Step 4: 225< 244 i.e. > c^{2} < a^{2} + b^{2}, and so the triangle is an acuteangle triangle.
The triangle is an acute angle triangle, as the final answer.
Example 4: Given Determine whether a triangle with sides
7 cm, 24 cm and 25 cm is an acuteangle, righttriangle or obtuseangle triangle.
Say the triangle is with sides a, b and c. Out of this
c is the longest side.
Step 1: In the given dimensions, assume a = 7, b = 24 and c = 25 cm
Step 2: a^{2} + b^{2} = 7^{2} + 24^{2} = 49 + 576 = 625
Step 3: Given, c = 25 i.e. c^{2} = 25^{2} = 625.
Step 4: 625 = 625 i.e. > c^{2} = a^{2} + b^{2},
and so the triangle is a right triangle.
The video above will explain more in detail about Pythagorean Theorem Converse,
and how to apply the concepts in solving realworld problems. This is explained
with the help of several examples and done watching video. This helps you to deal with solving problems and help doing the Geometry home work.
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