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## How to Solve Problems on Pythagorean Theorem?

Follow the simple steps listed here to solve problems related to the Pythagorean Theorem. They are along the lines

Step 1: Look at all the terms in the final equation

Step 2: Find out which right triangles contain those terms

Step 3: Start with those right triangles and apply the Pythagorean Theorem

## Pythagorean Theorem Word Problems

**Problem 1:**

A 35-foot ladder is leaning against the side of a building and its positioned such that the base of the ladder is 21 feet from the base of the building. Find the distance above the ground where the point of the ladder touches the building?

**Solution:**

Let the point above the ground = x

As given in the question,

Length of the foot ladder = 35

Base of the ladder = 21 feet

Let the equation be aÂ² + bÂ² = cÂ²

(21)Â² + (x)Â² = (35)Â²

441 + x = 1225

Subtracting by 441 on both sides

x = 784

Squaring and both sides

x = âˆš784

x = 28

28 feet far above the ground is the point where the ladder touches the building.

**Problem 2:**

Lainey runs a string of lights from the ground straight up a door frame that is 2.5 meters tall. Then they run the rest of the string in a straight line to a point on the ground that is 6 meters from the base of the door frame. There are 10 lights per meter of a string. How many total lights are on the string?

**Solution:**

As given in the question,

The length of the lights = 2.5 meters

The base of the door frame = 6 meters

No of lights per meter = 10

To find the total number of lights on a string, we write the equation

hÂ² = (2.5)Â² + (6)Â²

hÂ²= 6.25 + 36

hÂ²= 42.25

h = âˆš42.25

String Length = 2.5 + âˆš42.25

Therefore, no of lights = 9m * 10 lights = 90 lights

Thus, the total lights on the string = 90 lights

**Problem 3:**

Fencing at a hardware store costs $16.97 per yard. How much would it cost to fence in a triangular-shaped yard that has a leg of 11 feet and another leg of 60 feet?

Note: The yard is in the shape of a right triangle.

**Solution:**

Amount of fencing at hardware store = $16.97 per yard

Length of one leg = 11 feet

Length of another leg = 60 feet

As given the yard is in the shape of right triangle

We know the equation,

aÂ² + bÂ² = cÂ²

(11)Â² + (60)Â² = cÂ²

121 + 3600 = cÂ²

3721 = cÂ²

Squaring on both sides

âˆš3721 = âˆšcÂ²

61 = c

The amount for a fence in a triangular-shaped yard = $61

Thus, the final solution is $61

**Problem 4:**

If a 34-foot ladder is placed against the top of a 30-foot building. How many feet will be at the bottom of the ladder from the bottom of the building?

**Solution:**

As given in the question,

Length of the ladder = 34 foot

Length of the building = 30 foot

Let the length of bottom of the ladder = x

aÂ² + bÂ² = cÂ²

(30)Â² + xÂ² = (34)Â²

900 + xÂ² = 1156

xÂ² = 256

Add square root on both sides

âˆšxÂ² = âˆš256

x = 16 feets

Therefore, the bottom of the ladder from the bottom of the building will be 16 feets

Thus, the final solution is 16 feet

**Problem 5:**

A certain television is advertised as a 34-inch TV. If the width of the TV is 30 inches, how many inches tall is the TV?

**Solution:**

As given in the question,

Size of the TV = 34 inch

Width of the TV = 30 inch

Length of TV = x

To know the length of TV, we have to find the equation

aÂ² + bÂ² = cÂ²

(30)Â² + xÂ² = (34)Â²

900 + xÂ² = 1156

Add square root on both sides

âˆšxÂ² = âˆš1156

x = 34

TV is 34 inches tall.

Thus, the final solution is 34 inches

**Problem 6:**

Ramu starts driving north for 9 miles, then takes a right turn, and then he drives east for another 40 miles. At the end of driving, what is the distance of a straight line from the starting point?

**Solution:**

As given in the question,

Driving for north = 9 miles

Driving for east =40 miles

Let the straight line distance = x

The equation will be

aÂ² + bÂ² = cÂ²

(9)Â² + (40)Â² = cÂ²

81 + 1600 = cÂ²

1681 = cÂ²

Add square roots on both sides

âˆš1681 = âˆšcÂ²

41 = c

Therefore, the straight line distance from the starting point is 41 miles

Thus, the final solution is 41 miles

**Problem 7:**

In a right-angled triangle, the hypotenuse square is equal to the sum of the squares of other two sides?

**Solution:**

**Given:** A right-angle triangle in which one of the sides is 90Â°

**To prove:** ACÂ² = ABÂ² + BCÂ²

**Construction:** Draw BD âŠ¥ AC

In Î”’s ADB and ABC

âˆ ADB = âˆ ABC (Each 90Â°)

âˆ A = âˆ A (Common)

Î”ADB âˆ¼ Î”ABC (By A-A Criteria)

AD/AB = AB/AC

ABÂ² = AD * AC is the first equation

In Î”’s BDC and ABC

âˆ CDB = âˆ ABC (Each 90Â°)

âˆ C = âˆ C (Common)

Î”BDC âˆ¼ Î”ABC (By A-A Criteria)

DC/BC = BC/AC

BCÂ² = AC * DC is the (2) equation

ABÂ² = AD * AC is the (3) equation

Adding the equations (1) and (2)

ABÂ² + BCÂ² = AD * AC + AC * AD

ABÂ² + BCÂ² = AC (AD + DC)

ABÂ² + BCÂ² = AC (AC)

ABÂ² + BCÂ² = ACÂ²

âˆ´ Hence proved

**Problem 8:**

In the triangle, if the square of one side of the triangle is equal to the sum of the squares of another two sides of the triangle, then the angle that is opposite to the first side is a right angle triangle?

**Solution:**

**Â Given:** Î”ABC such that ABÂ² + BCÂ² = ACÂ²

**To prove:** âˆ B = 90Â°

**Construction:** Draw Î”DEF such that DE = AB, EF = BC and âˆ E = 90Â°

Since Î”DEF is right-angled,

By Pythagoras theorem

DEÂ² + EFÂ² = DFÂ²

ABÂ² + BCÂ² = DFÂ² (By const DE = AB and EF = BC)

ACÂ² = DFÂ² (Given ABÂ² + BCÂ² = ACÂ²)

AC = DF

**To prove:** âˆ B = 90Â°

In Î”s ABC and DEF

AB = DE (By const)

BC = EF (By const)

AC = DF (Proved)

Î”ABC â‰… Î”DEF (SSS)

âˆ B = âˆ E = 90Â°

Î”ABC is a right triangle

**Problem 9:**

Mary wants to cut across a rectangular lot rather than walk around it. Of the lot is 120 feet long and 50 feet wide. Mary walks diagonally across the lot, how many feet is the short cut?

**Solution:**

As given in the question,

Length of the lot = 120 feet

Width of the lot = 50 feet

Let the shortcut distance be x

As per the Pythagorean theorem,

aÂ² + bÂ² = cÂ²

(50)Â² + (120)Â² = xÂ²

2500 + 14,400 = xÂ²

16,900 = xÂ²

130 = x

Therefore, Mary walks 130 feet shortcut

Thus, the final solution is 130 feet

**Problem 10:**

The length of a living room is 2 feet less than twice its width. If the diagonal is 2 feet more than twice the width, find the dimensions of the room?

**Solution:**

Let the width of the living room = x

Given that

Length of the living room = 2 feet less than twice its width

Diagonal = 2 feet more than twice the width

From the given equations,

Width represents xÂ² + (2x – 2)Â² = (2x + 2)Â²

Length represents xÂ² = (2x + 2)Â² – (2x-2)Â²

From both the equations,

xÂ² = (4x + 8x + 4) – (4x – 8x + 4)

xÂ² = 16x

(xÂ²-16x) = 0

x(x-16) = 0

x = 0, x=16

Therefore, the width = 16

Length = (2x – 2) = 2(16) – 2 = 30

Thus, the length of the living room = 30 feet

Width = 16 feet

Hence, the final solution is l = 30 feet and w = 16 feet

**Problem 11:**

A man goes 12m east and 9m north. Find the distance from the initial point?

**Solution:**

As given in the question,

The distance man goes to east = 12m

i.e., BC = 12m

The distance man goes to north = 9m

i.e., AB = 9m

To find the initial point (AC), apply the Pythagorean theorem

ACÂ² = ABÂ² + BCÂ²

ACÂ² = (9)Â² + (12)Â²

ACÂ² = 81 + 144

ACÂ² = 225

AC = âˆš225

AC = 15

Therefore, he is 15m far from his initial point

Thus, the final solution is 15m

**Problem 12:**

Hari wants to hang a 7m long banner from the roof of her shop. The hooks for the strings are 10m apart. Hari wants the top of the banner to hang 1m below the roof. How long should each of the strings be?

**Solution:**

As per the question,

Length of the long banner = 7m

Hooks for strings = 10m apart

Length where the banner to hang below the roof = 1m

Let the length of the string = x

Hence, the equation will be

7 + 2x = 10

2x = 3

x = 1.5

Now, apply the Pythagorean theorem,

CÂ² = (1)Â² + (1.5)Â²

CÂ² = 1 + 2.25

CÂ² = 3.25

C = âˆš3.25

C = 1.80

Hence, each string should be 1.80m

Thus, the final solution is 1.80m

**Problem 13:**

Two kids are flying a kite with a string of 50 meters long. If the kids are 35 meters apart, how high is the kite off the ground?

**Solution:**

As given in the question,

Length of the string = 50 meters

The distance of kids apart = 35 meters

The height of the kite off the ground = b

Applying the Pythagorean theorem, we get

aÂ² + bÂ² = cÂ²

(35)Â² + bÂ² = (50)Â²

1225 + bÂ² = 2500

bÂ² = 1275

b = âˆš1275

b = 35.70

Therefore, 35.70mÂ is the kite off the ground

Thus, the final solution is 35.70 m

**Problem 14:**

A carpenter needs to add 2 braces to a barn door. If the door measures 12 * 16 feet, how much wood will he need for both braces?

**Solution:**

As given in the question,

No of braces = 2

Length of the door = 12 feet

The breadth of the door = 16 feet

To find the amount of wood for braces, we apply Pythagorean theorem

aÂ² + bÂ² = cÂ²

(12)Â² + (16)Â² = cÂ²

144 + 256 = cÂ²

400 = cÂ²

c = âˆš400

c = 20

Amount of wood for both the braces = 2 * 20 = 40ft

Therefore, the carpenter requires 40 ft of wood for both the braces.

Thus, the final solution is 40 ft

**Problem 15:**

M and N are points on the sides RP and RQ respectively of triangle PQR right-angled at R. Prove that PNÂ² + QMÂ² = PQÂ² + MNÂ²?

**Solution:**

PNÂ² = PRÂ² + RNÂ² is the (1) equation

QMÂ² = MRÂ² + RQÂ² is the (2) equation

Adding the equation (1) and (2)

PNÂ² + QMÂ² = MRÂ² + RQÂ² + PRÂ² + RNÂ²

PNÂ² + QMÂ² = (MRÂ² + RNÂ²) + (RQÂ² + PRÂ²)

PNÂ² + QMÂ² = MNÂ² + PQÂ²

âˆ´ Hence, it is proved