USING THE PYTHAGOREAN THEOREM

We can use the Pythagorean Theorem to find the length of a side of a right triangle when we know the lengths of the other two sides.

The Pythagorean Theorem

In a right triangle, the sum of the squares of the lengths of the  legs is equal to the square of the length of the hypotenuse.

If a and b are legs and c is the hypotenuse, then

a² + b²  =  c²

Problem 1 :

In the right triangle given below, find the length of the missing side using Pythagorean theorem. 

Solution :

Step 1 :

If a and b are legs and c is the hypotenuse, write Pythagorean for the above right triangle

a² + b²  =  c²

Step 2 :

Substitute the given measures.

7² + 24²  =  c²

Step 3 :

Solve the equation for c.

7² + 24²  =  c²

Simplify.

49 + 576  =  c²

625  =  c²

Write 625 as a perfect square (625  =  25²). 

25²  =  c²

Get rid of the square on both sides. 

25  =  c

Hence, the length of the hypotenuse is 25 inches.

Problem 2 :

In the right triangle given below, find the length of the missing side using Pythagorean theorem. 

Solution :

Step 1 :

If a and b are legs and c is the hypotenuse, write Pythagorean for the above right triangle

a² + b²  =  c²

Step 2 :

Substitute the given measures.

a² + 12²  =  15²

Step 3 :

Solve the equation for c.

Simplify.

a² + 144  =  225

Subtract 144 from both sides. 

a²  =  81

Write 81 as a perfect square (81  =  9²). 

a²  =  9²

Get rid of the square on both sides. 

a  =  9

Hence, the length of the leg is 9 centimeters.

Problem 3 : 

In the right triangle given below, find the length of the missing side using Pythagorean theorem. 

Solution :

Step 1 :

If a and b are legs and c is the hypotenuse, write Pythagorean for the above right triangle

a² + b²  =  c²

Step 2 :

Substitute the given measures.

30² + 40²  =  c²

Step 3 :

Solve the equation for c.

30² + 40²  =  c²

Simplify.

900 + 1600  =  c²

2500  =  c²

Write 2500 as a perfect square (2500  =  50²). 

50²  =  c²

Get rid of the square on both sides. 

50  =  c

Hence, the length of the hypotenuse is 50 ft.

Problem 4 : 

In the right triangle given below, find the length of the missing side using Pythagorean theorem. 

Solution :

Step 1 :

If a and b are legs and c is the hypotenuse, write Pythagorean for the above right triangle

a² + b²  =  c²

Step 2 :

Substitute the given measures.

a² + 40²  =  41²

Step 3 :

Solve the equation for c.

Simplify.

a² + 1600  =  1681

Subtract 1600 from both sides. 

a²  =  81

Write 81 as a perfect square (81  =  9²). 

a²  =  9²

Get rid of the square on both sides. 

a  =  9

Hence, the length of the leg is 9 inches.

Problem 5 : 

A wooden lagpole is 25 foot tall. In a storm, the lagpole is broken and its top touches the ground 5 foot from the base. Find the lengths of the segments of the lagpole.

Solution :

From the given information, it is clear that

AB = 25 foot, BC = 5 foot

AC² = AB² + BC²

AC² = 25² + 5²

= 625 + 25

AC² = 650

AC = √650

= 25.49

Approximating it as 26. So, the length of flagpole is 26 ft.

Problem 6 : 

Stanley has drawn a right angle triangle. One side is 14 cm and another is 18 cm. There are two possible lengths for the third side. What are they ?

Solution :

By considering the hypotenuse which measures 18 cm and by assuming the unknown side as x, we get

18² = x² + 14²

324 = x² + 196

324 - 196 = x²

x² = 128

x = √128

x = 11.31 cm

Approximately 11 cm.

By considering the hypotenuse as x, we get

x² = 18² + 14²

x² = 324 + 196

= 520

x = √520

= 22.8

Approximately 23 cm.

Problem 7 : 

ABC and BCD are right angle triangles. Find the length of AB

Solution :

In the triangle ABC,

BC² = AC² + AB²

BC² = 5² + x²

In the triangle BDC,

DB² = DC² + BC²

19² = 13² + BC²

361 - 169 = BC²

 BC² = 192

BC = √192

BC = 13.85

By applying the value of BC², we get

192 = 5² + x²

192 - 25 = x²

x² = 167

= 12.9

Approximately 13 cm.

Problem 8 : 

In the triangle ABC, AB = AC = x, BC = 10 cm and the area of the triangle 60 cm². Find x.

Solution :

BC = hypotenuse

Area of the triangle = (1/2) 10 (x)

60 = 5x

x = 60/5

x = 12

Problem 9 : 

An airplane is lying from Redville to Leek. The airplane lies 50 miles East and then 180 miles South. How far is Leek from Redville directly?

Solution :

AB = 50 miles, AC = 180 miles

BC² = AB² + AC²

BC² = 50² + 180²

= 2500 + 32400

= 34900

BC = √34900

= 186.81

= 187

So, the distance between Leek from Redville directly is 187 miles.

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