Mastering the Pythagorean Theorem: Calculating Hypotenuse in Right Triangles
Learn how to use the Pythagorean theorem to calculate the hypotenuse in right triangles effortlessly. Discover step-by-step methods, examples, and real-life applications of this fundamental mathematical concept.
These are examples of Pythagorean triangles with their hypotenuse calculated.
- Example 1:
- Given: Side lengths of 30 mm and 40 mm.
- Hypotenuse = √(30 mm)^2 + (40 mm)^2) = √(900 + 1600) = √2500 = 50 mm.
- Example 2:
- Given: Side lengths of 50 cm and 120 cm.
- Hypotenuse = √(50 cm)^2 + (120 cm)^2) = √(2500 + 14400) = √16900 = 130 cm.
- Example 3:
- Given: Side lengths of 70 dm and 240 dm.
- Hypotenuse = √(70 dm)^2 + (240 dm)^2) = √(4900 + 57600) = √62500 = 250 dm.
- Example 4:
- Given: Side lengths of 0.8 m and 1.5 m.
- Hypotenuse = √(0.8 m)^2 + (1.5 m)^2) = √(0.64 + 2.25) = √2.89 = 1.7 m.
- Example 5:
- Given: Side lengths of 90 m and 400 m.
- Hypotenuse = √(90 m)^2 + (400 m)^2) = √(8100 + 160000) = √168100 = 410 m.
- Example 6:
- Given: Side lengths of 120 m and 350 m.
- Hypotenuse = √(120 m)^2 + (350 m)^2) = √(14400 + 122500) = √136900 = 370 m.
- Example 7:
- Given: Side lengths of 150 m and 360 m.
- Hypotenuse = √(150 m)^2 + (360 m)^2) = √(22500 + 129600) = √152100 = 390 m.
- Example 8:
- Given: Side lengths of 0.16 km and 0.63 km.
- Hypotenuse = √(0.16 km)^2 + (0.63 km)^2) = √(0.0256 + 0.3969) = √0.4225 = 0.65 km.
- Example 9:
- Given: Side lengths of 2 m and 2.1 m.
- Hypotenuse = √(2 m)^2 + (2.1 m)^2) = √(4 + 4.41) = √8.41 = 2.9 m.
- Example 10:
- Given: Side lengths of 240 mm and 450 mm.
- Hypotenuse = √(240 mm)^2 + (450 mm)^2) = √(57600 + 202500) = √260100 = 510 mm.
- Example 11:
- Given: Side lengths of 280 cm and 450 cm.
- Hypotenuse = √(280 cm)^2 + (450 cm)^2) = √(78400 + 202500) = √280900 = 530 cm.
- Example 12:
- Given: Side lengths of 30 dm and 72 dm.
- Hypotenuse = √(30 dm)^2 + (72 dm)^2) = √(900 + 5184) = √6084 = 78 dm.
- Example 13:
- Given: Side lengths of 360 m and 770 m.
- Hypotenuse = √(360 m)^2 + (770 m)^2) = √(129600 + 592900) = √722500 = 850 m.
- Example 14:
- Given: Side lengths of 40 dm and 75 dm.
- Hypotenuse = √(40 dm)^2 + (75 dm)^2) = √(1600 + 5625) = √7225 = 85 dm.
- Example 15:
- Given: Side lengths of 4.8 km and 5.5 km.
- Hypotenuse = √(4.8 km)^2 + (5.5 km)^2) = √(23.04 + 30.25) = √53.29 = 7.3 km.
Pythagorean theorem and calculating the hypotenuse:
- What is the Pythagorean theorem?
- The Pythagorean theorem states that in a right-angled triangle, the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the lengths of the other two sides.
- How do I use the Pythagorean theorem to find the hypotenuse?
- To find the length of the hypotenuse, square the lengths of the other two sides, add them together, and then take the square root of the result.
- Can the Pythagorean theorem be used for any triangle?
- No, the Pythagorean theorem only applies to right-angled triangles, where one angle measures 90 degrees.
- What are the two shorter sides of a right triangle called?
- The two shorter sides of a right triangle are called the legs. The side opposite the right angle is called the hypotenuse.
- Do the units of measurement matter when using the Pythagorean theorem?
- Yes, the units of measurement must be consistent when using the Pythagorean theorem. Ensure that all side lengths are measured in the same units before performing calculations.
- Can I use fractions or decimals in the Pythagorean theorem?
- Yes, the Pythagorean theorem can be used with fractions, decimals, or whole numbers as long as the units are consistent.
- What if I only know the lengths of the legs of a right triangle?
- You can still use the Pythagorean theorem to find the length of the hypotenuse by squaring the lengths of the legs, adding them together, and then taking the square root of the result.
- What if I only know the length of the hypotenuse?
- If you know the lengths of one leg and the hypotenuse, you can use the Pythagorean theorem to find the length of the other leg by rearranging the formula.
- Are there any real-life applications of the Pythagorean theorem?
- Yes, the Pythagorean theorem is used in various fields such as architecture, engineering, navigation, and physics to solve problems involving distances, heights, and diagonals.
- Can the Pythagorean theorem be extended to three-dimensional shapes?
- Yes, the Pythagorean theorem can be extended to three-dimensional shapes, such as rectangular prisms, by considering the diagonals of faces and space diagonals.
- In a right triangle, the side opposite the right angle is called the __________.
- a) Leg
- b) Hypotenuse
- c) Adjacent side
- d) Opposite side
- The Pythagorean theorem states that in a right triangle, the square of the length of the __________ equals the sum of the squares of the lengths of the other two sides.
- a) Longer leg
- b) Shorter leg
- c) Hypotenuse
- d) Perpendicular side
- What is the formula for the Pythagorean theorem?
- a) �2+�2=�2
- b) �2=�2−�2
- c) �=�+�
- d) �=�2+�2
- If the lengths of the legs of a right triangle are 3 units and 4 units, what is the length of the hypotenuse?
- a) 6 units
- b) 7 units
- c) 8 units
- d) 5 units
- In a right triangle with legs of lengths 5 cm and 12 cm, what is the length of the hypotenuse?
- a) 13 cm
- b) 14 cm
- c) 15 cm
- d) 16 cm
- The Pythagorean theorem is only applicable to __________ triangles.
- a) Acute
- b) Obtuse
- c) Equilateral
- d) Right-angled
- Which term describes the two shorter sides of a right triangle?
- a) Legs
- b) Hypotenuse
- c) Base
- d) Altitude
- If the lengths of the legs of a right triangle are 6 units and 8 units, what is the length of the hypotenuse?
- a) 10 units
- b) 11 units
- c) 12 units
- d) 13 units
- If the length of one leg of a right triangle is 15 meters and the length of the hypotenuse is 17 meters, what is the length of the other leg?
- a) 8 meters
- b) 9 meters
- c) 10 meters
- d) 11 meters
- The units of measurement must be __________ when using the Pythagorean theorem.
- a) Different
- b) Random
- c) Inconsistent
- d) Consistent
- If the lengths of the legs of a right triangle are 9 inches and 12 inches, what is the length of the hypotenuse?
- a) 15 inches
- b) 16 inches
- c) 17 inches
- d) 18 inches
- Which side of a right triangle is directly opposite the right angle?
- a) Hypotenuse
- b) Shorter leg
- c) Longer leg
- d) Adjacent side
- The Pythagorean theorem is often used in fields such as __________.
- a) Mathematics only
- b) Medicine
- c) Architecture
- d) Literature
- If the lengths of the legs of a right triangle are 7 cm and 24 cm, what is the length of the hypotenuse?
- a) 23 cm
- b) 25 cm
- c) 26 cm
- d) 27 cm
- What is the first step in using the Pythagorean theorem to find the length of the hypotenuse?
- a) Square the lengths of the legs
- b) Find the difference between the lengths of the legs
- c) Add the lengths of the legs
- d) Take the square root of the sum of squares of the legs