What is Converse of Pythagorean Theorem?
Definition: The reverse of the Pythagorean Theorem states that we may determine whether a triangle is right-angled by comparing the sum of the squares of its two sides to the square of its third longer side.
To recap, the Pythagorean Theorem is a well-known theorem that lets us determine the length of the sides of a right triangle. It states that the sum of the squares of the two sides of a right triangle is equal to the square of the hypotenuse.
For instance, in the triangle given below, one of the sides is 6cm and the other is 8cm. The length of the hypotenuse is 10cm. As per the Pythagorean Theorem, the following holds true.
AB² + BC² = AC²
6² + 8² = 10²
36 + 64 = 100
The inverse of the Pythagorean Theorem asserts that if the square of a triangle’s longest side is equal to the sum of the squares of its two shorter sides, the triangle is a right triangle. In other words, the converse of the Pythagorean Theorem is the same theorem, only flipped. It allows us to quickly determine whether a triangle is a right triangle.
In connection to the theory given above, the three sides that fulfill the Pythagorean Theorem are called a Pythagorean triplet. Let’s take a close look at the sets of triangles given below.
Triangle 1 is a right triangle, because:
Applying the Pythagorean formula to the three sides of these triangles we can observe the following;
3² + 4² = 5²
9 + 16 = 25
25 = 25
Triangle 2 is an obtuse triangle, as one angle is greater than 90 degrees. Observe what happens when we apply the Pythagorean Theorem here.
5² + 5² < 8²
The sum of the squares of the two sides is lesser than 64
25 + 25 < 64
50 < 64
Now, observe when the formula is applied to the sides of an acute triangle. An acute triangle has angles whose measures are less than 90 degrees.
4² + 6² > 5.5²
The sum of the squares of the two sides is greater than that of the third side.
16 + 36 > 30.25
52 > 30.25
So this is proof that the Pythagorean Theorem is only applicable to right angled triangles.
Solved Converse of Pythagorean Theorem Examples
Check if the following sets of sides form the sides of a right triangle.
Example 1:
1. 2, 6, and 9.
Answer:
AB² + BC² = AC²
2² + 6² = 9²
4 + 36 < 81
40 < 81
The given set of sides does not form the sides of a right triangle.
Example 2:
2. 9, 12, and 15
Answer:
AB² + BC² = AC²
9² + 12² = 15²
81 + 144 = 225
225 = 225
The given set of sides form the sides of a right triangle.
Example 3:
If the points plotted below belong to a right triangle, find the length of the hypotenuse AC.
Answer:
According to the Pythagorean theorem
AB = 20 – 2 = 18 units
BC = 13 – 1 = 12 units
AC = ?
AB² + BC² = AC²
18² + 12² = AC²
324 + 144 = AC²
468 = AC²
AB² + BC² = AC = \(\sqrt{468}\) = 21.63
Therefore, the length of the hypotenuse is 21.63 units
Example 4:
A fountain is to be constructed near a church at the intersection of two walls in the shape of a right-triangle. If 5, 12 and 13 feet are the three proposed sides of the triangle, check if these sides are according to the requirements of the plan.
Solution:
In simple words, we have to verify if the three sides are those of a right triangle.
According to the Pythagorean theorem
AB² + BC² = AC²
5² + 12² = 13²
25 + 144 = 169
169 = 169
So, the three sides form the sides of a right triangle and this means it will be possible to create the structure.
Example 5:
Check if the triangle formed by the coordinates given below is a right triangle.
To find out if the given points form a Pythagorean triplet, we need to find the lengths of the sides AB, AC and BC. By looking at the graph, we can see that the length of BC = 25 – 0 = 25 units.
Next, draw a vertical line from point A meeting the x-axis at D(10,0). Mark a point E(10,25) and draw a line from C to E and connect E to A. ∠ADB and ∠AEC are right angles hence triangle ADB and triangle AEC are right triangles, finding their hypotenuses will be the length of the sides AC and AB.
AE = 25 – 5 = 20 units
EC = 20 – 10 = 10 units
AE² + EC² = AC²
20² + 10² = AC²
400 + 100 = AC²
500 = AC²
DB = 20 – 10 = 10 units
AD = 5 – 0 = 5 units
DB² + AD² = AB²
5² + 10² = AB²
25 + 100 = AB²
125 = AB²
To find if the three points A, B and C are a Pythagorean triplet, the following condition needs to be met.
AB² + AC² = BC²
500 + 125 = 25²
625 = 625
This means that the given points form a Pythagorean triplet, that is triangle ABC is a right triangle.
If a triangle is right-angled (one angle of the triangle is 90 degrees), the square of the hypotenuse equals the sum of the squares of the other two sides, according to Pythagoras’ theorem.
The reverse of the Pythagorean Theorem states that we may determine whether a triangle is acute, right, or obtuse by comparing the sum of the squares of its two sides to the square of its third side.
- The triangle is an acute triangle if the sum of the squares of the shorter sides is greater than the square of the longest side.
- The triangle is an obtuse triangle if the sum of the squares of the shorter sides is less than the square of the longest side.
- The triangle is a right triangle if the sum of squares of the shorter side is equal to the square of the longest side.