Worksheet: Real-Life Applications of the Pythagorean Theorem

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In this worksheet, we will practice applying the Pythagorean theorem in geometric questions and real-life situations.

Q1:

A window cleaner has a ladder that is 8.1 metres long. If he places it on the ground such that its top is at a window that is 6.76 metres above the ground, determine the distance between the base of the ladder and the wall, and round the result to the nearest hundredth.

Q2:

Amir traveled north for 19 miles and then east for 13 miles. Determine, to the nearest tenth of a mile, how far he is from his starting point.

Q3:

A ladder of height 24 feet leaning against a wall reaches a window 19 feet above the ground. To the nearest tenth, how far from the building is the bottom of the ladder?

Q4:

A man on the top of a building wants to have a guy wire extended to a point on the ground 20 ft from the base of the building. To the nearest foot, how long does the wire have to be if the building is 50 ft tall?

Q5:

A dining table is 24 ft long and 12 ft wide. Determine, to the nearest tenth, how far it is from one corner to the diagonally opposite corner.

Q6:

Find the perimeter of 𝐴 𝐵 𝐶 𝐷 .

Q7:

Find the area of the square 𝐵 𝐸 𝐷 𝐶 .

Q8:

Determine, to the nearest tenth of an inch, the length of the diagonal of a door of length 88 inches and width 34 inches.

Q9:

Maged has a rectangular back yard. He measured one side of the yard and found it to be 85 feet and its diagonal to be 117 feet. Determine, to the nearest tenth of a foot, the length of the other side of his back yard.

Q10:

Sarah and Farida are communicating by walkie-talkie with a range of 42 feet. Reaching a 22 ft long bridge, Sarah waits while Farida crosses and then walks up the bank until she can no longer hear Sarah. How far up the river did she walk? Give your answer to one decimal place.

Q11:

A tree, which was growing perpendicular to the ground, was 3 m tall. It snapped at a point that was 1 m above the ground. The top part of the tree fell and hit the ground. However, at the point of the break, the tree remained connected. Find the distance between the base of the tree and the point where the top of the tree touches the ground.

  • A √ 5 m
  • B √ 3 m
  • C √ 5 m
  • D √ 3 m
  • E 1 m

Q12:

Determine the value of 𝑥 to the nearest tenth.

Q13:

The figure shows a 129 m long bridge on supports 𝑀 𝐶 , 𝑀 𝐷 attached at the midpoint 𝑀 . If 𝐴 𝐶 = 5 1 . 6 m , find the length of 𝑀 𝐶 to the nearest hundredth.

Q14:

A space station is orbiting Earth at an altitude of 498 miles above the surface. Assuming the radius of the Earth is 4 0 0 0 miles, determine, to the nearest mile, the distance between the space station and the farthest point on the Earth’s surface from which it could be seen.

Q15:

Determine the length of 𝐶 𝐵 , given that 𝐴 𝐷 𝐶 and 𝐴 𝐷 𝐵 are two right-angled triangles, 𝐴 𝐵 = 1 3 , 𝐴 𝐷 = 5 , and 𝐴 𝐶 = 1 3 .

Q16:

Ancient Egyptians created right triangles using ropes with 13 equally spaced knots. The first and last knots (knots 1 and 13) were fixed together on the floor to form one vertex of the triangle. At which other two knots could the rope be fixed to form a right angle at knot 1?

  • A at knots 3 and 10
  • B at knots 5 and 8
  • C at knots 6 and 7
  • D at knots 4 and 9
  • E at knots 2 and 11

Q17:

A zip line was installed at an outside playground. It is 15.5 m long and attached to two posts 15 m apart on a flat ground. Given that at the arrival post, the zip line must be 1.5 metres above the ground, at what height should it be attached to the other post? Give your answer correct to one decimal place.

Q18:

Yara is a set designer. She wants to use two wooden pieces, a square 𝐴 𝐵 𝐶 𝐷 and a right triangle 𝐴 𝐷 𝐸 , which are attached together along 𝐴 𝐷 by hinges. She wants to place them on the floor, as shown in the figure, and make sure that the two pieces form a right angle. If this is the case, then any line in the triangle is perpendicular to 𝐴 𝐵 , and any line in the square is perpendicular to 𝐴 𝐸 . She has no tool with her except a measuring tape.

She measured the length of 𝐸 𝐵 and found it to be 42 cm. Is the angle between the square and the triangle a right angle?

  • Ano
  • Byes

Given that 𝐵 𝐶 is perpendicular to the floor, Norma could have measured the length of 𝐸 𝐶 to determine whether the two wooden pieces formed a right angle. What would the length of 𝐸 𝐶 have been, if it were a right angle?

  • A yes, 50.0 cm
  • B no, 40.3 cm
  • C yes, 70.3 cm
  • D yes, 58.3 cm
  • E no, 54.8 cm

Q19:

The distances between three cities are 77 miles, 36 miles, and 49 miles. Do the positions of these cities form a right triangle?

  • Ano
  • Byes

Q20:

Triangle 𝐴 𝐵 𝐶 has a right angle at 𝐵 . Given that 𝐴 𝐵 = 7 2 and 𝐵 𝐶 = 5 4 , find 𝐷 𝐹 and 𝐷 𝐸 , rounding your answers to the nearest hundredth.

  • A 𝐷 𝐹 = 4 1 . 8 3 , 𝐷 𝐸 = 6 4 . 4 0
  • B 𝐷 𝐹 = 4 8 . 3 0 , 𝐷 𝐸 = 5 5 . 7 7
  • C 𝐷 𝐹 = 4 0 . 5 0 , 𝐷 𝐸 = 9 6 . 0 0
  • D 𝐷 𝐹 = 2 5 . 9 2 , 𝐷 𝐸 = 3 4 . 5 6

Q21:

A right-angled triangle has side lengths of ( 2 𝑥 ) cm, ( 𝑥 + 5 ) cm, and ( 2 𝑥 + 4 ) cm. Find the value of 𝑥 and calculate the perimeter and the area of the triangle.

  • A 𝑥 = 4 , 29 cm, 36 cm2
  • B 𝑥 = 3 , 24 cm, 48 cm2
  • C 𝑥 = 4 , 29 cm, 72 cm2
  • D 𝑥 = 3 , 24 cm, 24 cm2
  • E 𝑥 = 1 , 14 cm, 6 cm2

Q22:

My empty garage is a rectangular prism that is 2 m high, and the dimensions of the floor are 3 m by 4 m. I have a ladder that is 5.2 m long. Will the ladder fit completely inside my garage?

  • Ayes
  • Bno

Q23:

In the trapezium 𝐴 𝐵 𝐶 𝐷 below, 𝐴 𝐷 ⫽ 𝐵 𝐶 and 𝐴 𝐸 ⟂ 𝐵 𝐶 . Find its area.

Q24:

Determine the area of the rectangle whose diagonal length is 55.1 cm, given that the length of one of its dimensions is 39.9 cm.

Q25:

Given that the area of the square 𝐴 𝐵 𝐹 𝐺 is 1 024 cm2, and the area of the square 𝐴 𝐶 𝐷 𝐸 is 1 600 cm2, find the side length of the shaded square.

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