**Pythagorean theorem word problems worksheet :**

Worksheet given in this section is much useful to the students who would like to practice solving word problems on "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

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.

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.

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.

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 ?

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 ?

**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 is

x - 17 = 24 - 17 = 7 inches

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

**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 worksheet".

**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 is

49 - x = 49 - 40 = 9

If x = 9, then the length of other side is

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 worksheet".

**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.

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 worksheet".

**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.

So, we have , 8² + 6² = x²

64 + 36 = x²

100 = x²

10² = x²

10 = x

Hence, the length of the ladder is 10 feet.

After having gone through the stuff given above, we hope that the students would have understood "Pythagorean theorem word problems worksheet".

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