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What is the Hypotenuse in Trigonometry?
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 90-degree angle. This special side is crucial for understanding trigonometry.
Simple Example
Quick Example
Imagine you are flying a kite. The string holding the kite up makes a straight line. If the kite is directly above a point on the ground and you are standing some distance away, the kite string forms the hypotenuse of a right-angled triangle, with the ground and the vertical height as the other two sides.
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 problem. We have a right-angled triangle and need to find the longest side (hypotenuse).
---Step 2: Recall the Pythagoras theorem, which 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 3: Assign the given side lengths. Let a = 3 cm and b = 4 cm.
---Step 4: Substitute these values into the Pythagoras theorem: 3^2 + 4^2 = c^2.
---Step 5: Calculate the squares: 9 + 16 = c^2.
---Step 6: Add the numbers: 25 = c^2.
---Step 7: To find 'c', take the square root of both sides: c = sqrt(25).
---Step 8: Calculate the square root: c = 5 cm.
Answer: The hypotenuse of the triangle is 5 cm.
Why It Matters
The hypotenuse is fundamental for calculating distances and angles, which is vital in many fields. Engineers use it to design stable buildings, while physicists use it to analyze forces and movements. It's also used by navigators to plot routes, helping pilots and ship captains reach their destinations safely.
Common Mistakes
MISTAKE: Confusing the hypotenuse with the adjacent or opposite side | CORRECTION: The hypotenuse is ALWAYS opposite the 90-degree angle and is the longest side.
MISTAKE: Forgetting to take the square root at the end when using the Pythagorean theorem | CORRECTION: After finding c^2, remember to take the square root to find the actual length of '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-angled triangle, if the two shorter sides are 6 units and 8 units, what is the length of the hypotenuse? | ANSWER: 10 units
QUESTION: A ladder 13 meters long leans against a wall. If the base of the ladder is 5 meters away from the wall, how high up the wall does the ladder reach? (Hint: The ladder is the hypotenuse) | ANSWER: 12 meters
QUESTION: A right-angled triangle has one side of length 7 cm and its hypotenuse is 25 cm. What is the length of the third side? | ANSWER: 24 cm
MCQ
Quick Quiz
Which of the following statements about the hypotenuse is correct?
It is always the shortest side of a right-angled triangle.
It is always opposite the smallest angle.
It is always opposite the 90-degree angle.
It is only found in equilateral triangles.
The Correct Answer Is:
C
The hypotenuse is by definition the side opposite the right (90-degree) angle, making it the longest side. Options A, B, and D are incorrect descriptions of the hypotenuse.
Real World Connection
In the Real World
When ISRO scientists design rockets or satellites, they use trigonometry, which relies on the hypotenuse, to calculate trajectories and ensure precise launches. Similarly, architects use these concepts to determine the slant height of roofs or the stability of ramps in buildings, making sure everything is safe and sound for us to use.
Key Vocabulary
Key Terms
RIGHT-ANGLED TRIANGLE: A triangle with one angle exactly 90 degrees | PYTHAGORAS THEOREM: A fundamental rule (a^2 + b^2 = c^2) used to find side lengths in right-angled triangles | ANGLE: The space between two intersecting lines or surfaces | SIDE: One of the lines that form a polygon
What's Next
What to Learn Next
Great job understanding the hypotenuse! Next, you should explore 'Trigonometric Ratios (Sine, Cosine, Tangent)'. These ratios use the hypotenuse to define relationships between angles and sides, helping you solve even more complex problems in triangles!
