The Pythagorean theorem

says that in any right triangle, the sum of the squares of the two shoter sides (legs) is equal to the square of of the longest side (hypotenuse).

If you know two sides of any right triangle, the Pyhagorean theorem can always be used to find the third side.

Example 1

In the figure below, what is x ?

9² +  x² =  15²

81 +  x² = 225

x² = 225

  x = 12

There is also

Converse of the Pythagorean theorem:

If  the square of the length of the longest side of a triangle is equal to the sum of the squares of the lengths of the other two sides, then the triangle is right triangle.

if  c² = a ² + b², then  ∆ ABC  is a right triangle.

Also you can use the modified Pythagorean theorem to find  whether a triangle is acute or obtuse:

1. 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 other two sides, then the triangle is acute.

if  c² < a ² + b², then  ∆ ABC  is acute triangle.

2. If the square of the length of the longest side of a triangle is greater than the sum of the squares of the lengths of other two sides, then the triangle is obtuse.

if  c² > a ² + b², then  ∆ ABC  is obtuse triangle.

 

Problem 1

In the fugure above, two right triangles share a common side. What is the length of  AB ?

Solution

Following The Pythagorean theorem, in ∆ BCD:

BD² = BC²  – CD² = 10²-6²= 100-36 = 64 ⤇ BD = 8

In ∆ BDA:

BA² = BD²  – AD² = 8²-6²= 64-36 = 28 ⤇ BA= √28 or 2√7

 Problem 2

In the figure above, two right triangles share a common side. If AB= 5, BD = 12, and CD = 3√3,

what is the length of AC ?

Solution

Following The Pythagorean theorem, in ∆ ABD:

AD² = BD² +AB² = 12²+5²= 144+25 = 169 ⤇ AD = 13

In ∆ ACD:

AC² = AD²  + CD² = 169+27= 196  ⤇ AC = 14