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What is the Hypotenuse of a Right Triangle?
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 directly opposite the right angle (the 90-degree angle).
Simple Example
Quick Example
Imagine a ladder leaning against a straight wall. The ground, the wall, and the ladder form a right-angled triangle. The ladder itself is the hypotenuse because it's the longest side and is opposite the corner where the wall meets the ground (the right angle).
Worked Example
Step-by-Step
Let's find the hypotenuse of a right triangle where the other two sides are 3 cm and 4 cm.
Step 1: Understand the Pythagorean Theorem. It states that in a right 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.
---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 numbers: 25 = c^2.
---Step 6: To find c, take the square root of both sides: c = sqrt(25).
---Step 7: Calculate the square root: c = 5.
The hypotenuse of the triangle is 5 cm.
Why It Matters
Understanding the hypotenuse is crucial in fields like engineering for designing structures, in physics for calculating forces, and in space technology for rocket trajectories. Architects, game developers, and even medical imaging specialists use this concept regularly.
Common Mistakes
MISTAKE: Confusing the hypotenuse with one of the shorter sides. | CORRECTION: Always remember the hypotenuse is the longest side and is opposite the 90-degree angle.
MISTAKE: Forgetting to take the square root at the end of the Pythagorean calculation, leaving the answer as c^2. | CORRECTION: After calculating a^2 + b^2, remember to take the square root of the sum to find the actual length of the hypotenuse (c).
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 right-angled triangles.
Practice Questions
Try It Yourself
QUESTION: In a right triangle, if the two shorter sides are 6 cm and 8 cm, what is the length of the hypotenuse? | ANSWER: 10 cm
QUESTION: A painter places a ladder 5 meters away from a wall. If the top of the ladder reaches 12 meters up the wall, how long is the ladder (the hypotenuse)? | ANSWER: 13 meters
QUESTION: The hypotenuse of a right triangle is 17 cm, and one of its shorter sides is 8 cm. Find the length of the other shorter side. | ANSWER: 15 cm
MCQ
Quick Quiz
Which statement about the hypotenuse of a right-angled triangle is correct?
It is always one of the two shortest sides.
It is always opposite the smallest angle.
It is always the side opposite the 90-degree angle.
It can be any side of the triangle.
The Correct Answer Is:
C
The hypotenuse is defined as the side opposite the right angle (90-degree angle) and is always the longest side. Options A, B, and D are incorrect based on this definition.
Real World Connection
In the Real World
When you use Google Maps or any navigation app, it often calculates the shortest distance 'as the crow flies' between two points. This 'straight line' distance can often be thought of as the hypotenuse in a right triangle formed by the north-south and east-west distances, helping delivery services like Zepto or Swiggy find optimal routes.
Key Vocabulary
Key Terms
RIGHT ANGLE: An angle that measures exactly 90 degrees. | PYTHAGOREAN THEOREM: A fundamental relation in Euclidean geometry among the three sides of a right triangle. | TRIANGLE: A polygon with three edges and three vertices. | LEG: The two shorter sides of a right triangle that form the right angle.
What's Next
What to Learn Next
Now that you understand the hypotenuse, you can explore the basics of trigonometry, like sine, cosine, and tangent. These concepts build directly on the hypotenuse and will help you solve even more complex problems involving angles and distances.
