# PYTHAGOREAN THEOREM WORD PROBLEMS

## About "Pythagorean theorem word problems"

Pythagorean theorem word problems play a key role in competitive exams.

To score more in competitive exams, we must be aware of "How to solve word problems on Pythagorean theorem".

## Pythagorean theorem

In any right angle triangle, square of the hypotenuse is equal to sum of the squares of other two sides.

That is,

a²    +    b²    =    c²

## Pythagorean theorem word problems

Now, let us look at some Pythagorean theorem word problems.

Problem 1 :

If the height of a triangle is 17 inches less than the length of its base and the length of the hypotenuse is 25 inches, find the base and the height.

Solution :

Let "x" be the length of the base.

Then, the height  =  x - 17

Hypotenuse  =  25

According to Pythagorean theorem, the square of the hypotenuse is equal to sum of the squares of other two sides.

So, we have ,  x² + (x-7)²   =   25²

x² + x² - 2(x)(17) + 17²  =  625

2x² - 34x + 289   =   625

2x² - 34x - 336   =   0

x² -17x - 168   =   0

(x+7) (x-24)   =  0

x = -7  or  x = 24

x = -7 can not be accepted. (Because length can never be negative)

Then the length of the base ----> x  =  24 inches

the length of the height ----> x - 17  =  24 - 17  =  7 inches

Let us look at the next problem on "Pythagorean theorem word problems"

Problem 2  :

The sides of an equilateral triangle are shortened by 12 units, 13 units, 14 units respectively and a right angle triangle is formed. Find the length of side of the equilateral triangle.

Solution :

Let "x" be the length of side of the equilateral side.

The sides of an equilateral triangle are shortened by 12 units, 13 units, 14 units and a right angle triangle is formed.

Then, new lengths are  (x-12), (x-13) and (x-14)

These are the sides of a right angle triangle.

And also hypotenuse = (x-12)

Because (x-12) is the longest side.

According to Pythagorean theorem, the square of the hypotenuse is equal to sum of the squares of other two sides.

So, we have ,  (x - 13)² + (x - 14)²   =   (x - 12)²

x² - 2(x)(13) + 13² + x² - 2(x)(14) + 14² = x² - 2(x)(12) + 144

x² - 26x + 169 + x² - 28x + 196 = x² - 24x + 144

2x² - 54x + 365  =  x² - 24x + 144

x² - 30x + 221   =   0

(x - 13)(x - 17)   =   0

x  =  13  or x  =  17

If we take x = 13, the sides of the right triangle are

x - 12 = 1,   x - 13 = 0,  x - 14 = -1

When x = 13, we get one of the sides is zero and the sign of the another side is negative.

So, x = 13 can not be accepted.

If we take x = 17, the sides of the right triangle are

x - 12 = 5,   x - 13 = 4 ,  x - 14 = 3

All of the three sides of the right angle are positive when x = 17.

So, x = 17 can be accepted.

Hence, the length of side of equilateral triangle is 17 units.

Let us look at the next problem on "Pythagorean theorem word problems"

Problem 3  :

If the sum of the lengths of two sides of a right triangle is 49 inches and the hypotenuse is 41 inches,then find the two sides.

Solution :

Let "x" be one of the two sides.

Given : Sum of the two sides = 49

Then the length of other side = 49 - x

And also hypotenuse = 41

According to Pythagorean theorem, the square of the hypotenuse is equal to sum of the squares of other two sides.

So, we have ,  x² + (49 - x)²   =   41²

x² + 49² - 2(49)(x) + x²  =  1681

2x² - 98x + 2401   =   1681

2x² - 98x + 720  =   0

x² - 49x + 360   =   0

(x-40) (x-9)   =  0

x = 40  or  x = 9

If x = 40, then the length of other side -----> 49 - x -----> 49 - 40 = 9

If x = 9, then the length of other side -----> 49 - x -----> 49 - 9 = 40

Hence, the lengths of other two sides are 9 inches and 40 inches.

Let us look at the next problem on "Pythagorean theorem word problems"

Problem 4  :

From a train station, one train heads north, and another heads east. Some time later, the northbound train has traveled 12 miles, and the eastbound train has traveled 16 miles. How far apart are the two trains, measured in a straight line?

Solution :

Let "x" be the required distance.

According to Pythagorean theorem, the square of the hypotenuse is equal to sum of the squares of other two sides.

So, we have ,  12² + 16²   =   x²

144 + 256  =  x²

400   =   x²

20²  =  x²

20  =  x

Hence, the distance between the two trains is 20 miles.

Let us look at the next problem on "Pythagorean theorem word problems"

Problem 5  :

The foot of a ladder is placed 6 feet from a wall. If the top of the ladder rests 8 feet up on the wall, how long is the ladder?

Solution :

Let "x" be the required distance.

According to Pythagorean theorem, the square of the hypotenuse is equal to sum of the squares of other two sides.

So, we have ,  8² + 6²   =   x²

64 + 36  =  x²

100   =   x²

10²  =  x²

10  =  x

Hence, the length of the ladder is 10 feet.

So far we have seen some practice questions on "Pythagorean theorem word problems".

Even though we can understand Pythagorean word problems, it is also important to understand the proof of Pythagorean theorem.

That, we are are going to see in the next section.

## Pythagorean theorem proof

To get proof of Pythagorean theorem, let us take the right triangle as given below.

Now, let us annex a square on each side of the triangle as given below.

(Size of each small box in the squares 1, 2 and 3 are same in size)

In square "1", each side is divided into 3 units equally.

Then the side length of square "1", a = 3

In square "2", each side is divided into 4 units equally.

Then the side length of square "2", b = 4

In square "3", each side is divided into 5 units equally.

Then the side length of square "3", c = 4

Area of three squares

Area of square "1"   ------>   a²   =   3²   =   9 square units

Area of square "2"   ------>   b²   =   4²   =   16 square units

Area of square "3"   ------>   c²    =   5²   =   25 square units

9    +    16    =    25  --------> (This is true)

Then, we have

Area of square "1"  +  Area of square "2"  =  Area of square "3"

a²    +    b²    =    c²

From the above result, in the given right triangle, it is very clear that the sum of squares of the sides "a" and "b" is equal to square of the third side "c".

Hence, Pythagorean theorem is proved.

Important points about right angle triangle :

1. The longest side is called as "hypotenuse"

2. In the above triangle "c" is hypotenuse.

3. The side which is opposite to right angle is hypotenuse.

4. Always the square of longest side (hypotenuse) is equal to the sum of the squares of other two sides.

5. If we know the lengths of two sides of a right angle triangle, we will be able to know the length of the third side using Pythagorean theorem.

After having gone through the stuff, we hope that the students would have understood "How to do Pythagorean theorem word problems".

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