What is the Hypotenuse?

S6-SA2-0002

Grade Level:

Class 10

AI/ML, Physics, Biotechnology, Space Technology, Chemistry, Engineering, Medicine

Definition

What is it?

The hypotenuse is the longest side of a right-angled triangle. It is always the side opposite the 90-degree angle. This special side is crucial for understanding the Pythagorean Theorem.

Simple Example

Quick Example

Imagine you are flying a kite. The string holding the kite is like the hypotenuse. The height of the kite above the ground and the distance of the kite from you on the ground form the other two sides of a right-angled triangle. The string is the longest path.

Worked Example

Step-by-Step

Let's find the hypotenuse of a right-angled triangle where the other two sides are 3 cm and 4 cm.
---Step 1: Understand the Pythagorean Theorem. It states that in a right-angled triangle, the square of the hypotenuse (c) is equal to the sum of the squares of the other two sides (a and b). So, a^2 + b^2 = c^2.
---Step 2: Identify the given sides. Here, a = 3 cm and b = 4 cm. We need to find c (the hypotenuse).
---Step 3: Substitute the values into the formula: 3^2 + 4^2 = c^2.
---Step 4: Calculate the squares: 9 + 16 = c^2.
---Step 5: Add the values: 25 = c^2.
---Step 6: To find c, take the square root of 25: c = sqrt(25).
---Step 7: Calculate the square root: c = 5.
---Answer: The hypotenuse of the triangle is 5 cm.

Why It Matters

Understanding the hypotenuse is fundamental in fields like Engineering, where architects use it to design stable buildings, and in Space Technology for calculating rocket trajectories. It's also vital for game developers in AI/ML to make objects move realistically on screen.

Common Mistakes

MISTAKE: Confusing the hypotenuse with any other side of a triangle. | CORRECTION: Always remember the hypotenuse is *only* in a right-angled triangle and is *always* opposite the 90-degree angle.

MISTAKE: Forgetting to take the square root at the end of a Pythagorean Theorem calculation, leaving the answer as c^2 instead of c. | CORRECTION: After calculating a^2 + b^2, remember to take the square root of the sum to find the actual length of the hypotenuse.

MISTAKE: Applying the Pythagorean Theorem to triangles that are not right-angled. | CORRECTION: The Pythagorean Theorem and the concept of a hypotenuse apply *only* to triangles that have one angle exactly 90 degrees.

Practice Questions

Try It Yourself

QUESTION: A ladder is 10 meters long and leans against a wall. The base of the ladder is 6 meters away from the wall. What is the height of the wall where the ladder touches it? (Hint: The ladder is the hypotenuse) | ANSWER: 8 meters

QUESTION: In a right-angled triangle, one side is 5 cm and the hypotenuse is 13 cm. What is the length of the third side? | ANSWER: 12 cm

QUESTION: An auto-rickshaw travels 8 km North and then 6 km East. How far is the auto-rickshaw from its starting point in a straight line? | ANSWER: 10 km

MCQ

Quick Quiz

Which statement about the hypotenuse is TRUE?

It is the shortest side of a right-angled triangle.

It is always opposite the smallest angle.

It is the side opposite the 90-degree angle in a right-angled triangle.

It can be found in any type of triangle.

The Correct Answer Is:

The hypotenuse is specifically defined as the longest side of a right-angled triangle, and it is always located opposite the 90-degree angle. Options A, B, and D are incorrect as they contradict this definition.

Real World Connection

In the Real World

When a civil engineer designs a ramp for a building or a flyover in a city like Mumbai, they use the hypotenuse to calculate the length of the ramp. Similarly, a carpenter building a roof often uses this concept to find the correct length of the slanting beams (rafters) for stability and safety.

Key Vocabulary

Key Terms

RIGHT-ANGLED TRIANGLE: A triangle with one 90-degree angle. | PYTHAGOREAN THEOREM: A fundamental relation in Euclidean geometry among the three sides of a right-angled triangle. | SQUARE ROOT: A number that, when multiplied by itself, equals a given number. | OPPOSITE SIDE: The side of a triangle that is across from a specific angle.

What's Next

What to Learn Next

Now that you understand the hypotenuse, you're ready to dive deeper into the Pythagorean Theorem and its applications! Next, you can explore how to use it to solve more complex problems involving distances and geometry in 3D shapes.