Lesson Worksheet: Applications on the Pythagorean Theorem Mathematics • 8th Grade

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


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


Benjamin 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.


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?


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.


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


Liam 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.


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


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.

  • A1 m
  • B5 m
  • C5 m
  • D3 m
  • E3 m


The figure shows a 129 m long bridge on supports 𝑀𝐶, 𝑀𝐷 attached at the midpoint 𝑀. If 𝐴𝐶=51.6m, find the length of 𝑀𝐶 to the nearest hundredth.


A space station is orbiting Earth at an altitude of 498 miles above the surface. Assuming the radius of the Earth is 4,000 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.


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?

  • Aat knots 6 and 7
  • Bat knots 5 and 8
  • Cat knots 2 and 11
  • Dat knots 4 and 9
  • Eat knots 3 and 10


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 meters above the ground, at what height should it be attached to the other post? Give your answer correct to one decimal place.


Olivia is a set designer. She wants to use two wooden pieces, a square 𝐴𝐵𝐶𝐷 and a right triangle 𝐴𝐷𝐸, which are attached together along 𝐴𝐷 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 𝐴𝐵, and any line in the square is perpendicular to 𝐴𝐸. She has no tool with her except a measuring tape.

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

  • AYes
  • BNo

Given that 𝐵𝐶 is perpendicular to the floor, Olivia could have measured the length of 𝐸𝐶 to determine whether the two wooden pieces formed a right angle. What would the length of 𝐸𝐶 have been if it were a right angle? Round your answer to the nearest tenth.


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


Triangle 𝐴𝐵𝐶 has a right angle at 𝐵. Given that 𝐴𝐵=72 and 𝐵𝐶=54, find 𝐷𝐹 and 𝐷𝐸, rounding your answers to the nearest hundredth.

  • A𝐷𝐹=40.50, 𝐷𝐸=96.00
  • B𝐷𝐹=48.30, 𝐷𝐸=55.77
  • C𝐷𝐹=25.92, 𝐷𝐸=34.56
  • D𝐷𝐹=41.83, 𝐷𝐸=64.40


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

  • A𝑥=4, 29 cm, 72 cm2
  • B𝑥=1, 14 cm, 6 cm2
  • C𝑥=3, 24 cm, 48 cm2
  • D𝑥=4, 29 cm, 36 cm2
  • E𝑥=3, 24 cm, 24 cm2


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?

  • Ano
  • Byes


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.


Consider the isosceles right triangle, 𝑇, shown in the figure. Squares have been added to each side of 𝑇.

Let the area of 𝑇 be one area unit. The areas of the squares on the legs are 𝐴 and 𝐴, and the area of the square on the hypotenuse is 𝐴. Find a relationship between the area of the square on the hypotenuse and the areas of the squares on the legs.

  • A𝐴=2(𝐴+𝐴)
  • B𝐴=12(𝐴+𝐴)
  • C𝐴=𝐴=𝐴
  • D𝐴=(𝐴+𝐴)
  • E𝐴=𝐴+𝐴


Find the area of the trapezoid 𝐴𝐵𝐶𝐷.


In the figure below, 𝐴𝐵=𝐵𝐶=𝐴𝐷=𝐷𝐸. Find the area of the shaded square.


In the figure below, 𝐴𝐵𝐶𝐷 is a parallelogram in which 𝐴𝐷=82cm and 𝐷𝐶=39cm. Point 𝐸 is on segment 𝐵𝐶 with 𝐶𝐸=31.2 and 𝐷𝐸=23.4. Find the area of 𝐴𝐵𝐶𝐷.


Ancient Egyptians are said to have used ropes to create right angles. The rope was divided into twelve equal lengths with 13 equally spaced knots. The two ends of the rope (knots 1 and 13) were fixed together on the floor. At which two knots should the rope be stretched tight to form a right angle at knot 1?

  • Aat knots 3 and 10
  • Bat knots 6 and 7
  • Cat knots 2 and 11
  • Dat knots 4 and 9
  • Eat knots 5 and 8


A carpenter wishes to construct a trapezoidal prism for the base of a table where the cross section is a regular trapezoid, as seen in the given figure. 𝐴𝐵 = 32 inches, 𝐷𝐶 = 20 inches, and the perpendicular distance between 𝐴𝐵 and 𝐷𝐶 is 24 inches.

Work out the length of 𝐴𝐷.

Work out the measure of angle 𝐴𝐷𝐶.

If the carpenter then decides to add a strut between 𝐴 and 𝐶, find the length of 𝐴𝐶.


𝐴𝐵𝐶𝐷 is a trapezoid-shaped piece of land where 𝐴𝐷 is parallel to 𝐵𝐶 and 𝐴𝐵𝐵𝐶. Find the length of 𝐴𝐵 given 𝐴𝐷=24m, 𝐵𝐶=48m and 𝐷𝐶=30m. Give the answer to the nearest meter.

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