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What is an Isosceles Right-Angled Triangle in Trigonometry?
Grade Level:
Class 10
AI/ML, Physics, Biotechnology, Space Technology, Chemistry, Engineering, Medicine
Definition
What is it?
An Isosceles Right-Angled Triangle is a special type of triangle that has two sides of equal length AND one angle that measures exactly 90 degrees. In trigonometry, we often use its unique angle properties (45-45-90 degrees) to solve problems.
Simple Example
Quick Example
Imagine you're cutting a square piece of paper diagonally from one corner to the opposite corner. Each of the two triangles formed will be an isosceles right-angled triangle. The two sides touching the 90-degree corner will be equal in length.
Worked Example
Step-by-Step
PROBLEM: In an isosceles right-angled triangle, if the two equal sides are 5 cm each, find the length of the hypotenuse.
---STEP 1: Identify the given information. We have an isosceles right-angled triangle. This means two sides are equal, and one angle is 90 degrees. The equal sides are 5 cm each.
---STEP 2: Recall the Pythagorean theorem for right-angled triangles: a^2 + b^2 = c^2, where 'a' and 'b' are the lengths of the two shorter sides (legs), and 'c' is the length of the hypotenuse.
---STEP 3: Substitute the given values into the theorem. Here, a = 5 cm and b = 5 cm. So, 5^2 + 5^2 = c^2.
---STEP 4: Calculate the squares: 25 + 25 = c^2.
---STEP 5: Add the values: 50 = c^2.
---STEP 6: To find 'c', take the square root of 50. c = sqrt(50).
---STEP 7: Simplify the square root: sqrt(50) = sqrt(25 * 2) = 5 * sqrt(2).
---ANSWER: The length of the hypotenuse is 5 * sqrt(2) cm (approximately 7.07 cm).
Why It Matters
Understanding these triangles helps engineers design stable structures and calculate forces in physics. In game development and AI, knowing these shapes helps in creating realistic environments and movements. It's a foundational concept for careers in architecture, robotics, and even space technology.
Common Mistakes
MISTAKE: Assuming all right-angled triangles are isosceles. | CORRECTION: Only a right-angled triangle with two equal sides (or two equal acute angles) is isosceles.
MISTAKE: Confusing the hypotenuse with one of the equal sides. | CORRECTION: The hypotenuse is always the longest side, opposite the 90-degree angle, and it is never equal to the other two sides in an isosceles right triangle.
MISTAKE: Incorrectly applying angle sum property, thinking all angles are 60 degrees. | CORRECTION: In an isosceles right-angled triangle, the angles are always 90, 45, and 45 degrees. The sum is 180 degrees.
Practice Questions
Try It Yourself
QUESTION: If the hypotenuse of an isosceles right-angled triangle is 10 cm, what is the length of each of the equal sides? | ANSWER: 5 * sqrt(2) cm (approx 7.07 cm)
QUESTION: An isosceles right-angled triangle has one angle as 90 degrees. What are the measures of the other two angles? | ANSWER: Both are 45 degrees.
QUESTION: A ladder leaning against a wall forms an isosceles right-angled triangle with the ground and the wall. If the base of the ladder is 6 meters from the wall, how long is the ladder? | ANSWER: 6 * sqrt(2) meters (approx 8.49 meters)
MCQ
Quick Quiz
Which of the following statements is TRUE for an isosceles right-angled triangle?
All three sides are equal.
Two sides are equal, and all angles are 60 degrees.
Two sides are equal, and one angle is 90 degrees.
No sides are equal, but one angle is 90 degrees.
The Correct Answer Is:
C
An isosceles triangle means two sides are equal, and a right-angled triangle means one angle is 90 degrees. Option C combines both definitions correctly.
Real World Connection
In the Real World
When architects design roof trusses for houses or build support structures for bridges, they often use isosceles right-angled triangles for stability and strength. Even the triangular safety signs you see on Indian roads sometimes use this shape for clear visibility and design efficiency.
Key Vocabulary
Key Terms
ISOSCELES: Having two sides of equal length | RIGHT-ANGLED TRIANGLE: A triangle with one 90-degree angle | HYPOTENUSE: The longest side of a right-angled triangle, opposite the 90-degree angle | PYTHAGOREAN THEOREM: a^2 + b^2 = c^2, relating the sides of a right triangle
What's Next
What to Learn Next
Great job understanding this special triangle! Next, explore basic trigonometric ratios (sine, cosine, tangent). Knowing the 45-45-90 properties of this triangle will make understanding those ratios much easier and more intuitive.
