What is the Ratio of Sides in a 30-60-90 Triangle?

S6-SA2-0138

Grade Level:

Class 10

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

Definition

What is it?

A 30-60-90 triangle is a special right-angled triangle where the angles are 30 degrees, 60 degrees, and 90 degrees. The sides of this triangle always follow a specific ratio, making it easy to find side lengths if you know just one side.

Simple Example

Quick Example

Imagine you're designing a ramp for a skateboard park that forms a 30-60-90 triangle with the ground. If the shortest side of the ramp (opposite the 30-degree angle) is 5 meters, then the hypotenuse (the ramp itself) will be 10 meters, and the base (opposite the 60-degree angle) will be 5 * sqrt(3) meters. You don't need to measure all sides!

Worked Example

Step-by-Step

Let's find the lengths of the other two sides of a 30-60-90 triangle if the side opposite the 60-degree angle is 9 cm.

Step 1: Understand the ratio. The sides of a 30-60-90 triangle are in the ratio 1 : sqrt(3) : 2, corresponding to the sides opposite the 30-degree, 60-degree, and 90-degree angles, respectively.

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Step 2: Identify the given side. We are given the side opposite the 60-degree angle is 9 cm. In our ratio, this corresponds to 'sqrt(3)'.

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Step 3: Find the 'unit' length (side opposite 30-degree angle). Let 'x' be the side opposite the 30-degree angle. Then, the side opposite the 60-degree angle is x * sqrt(3). So, x * sqrt(3) = 9.

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Step 4: Solve for x. x = 9 / sqrt(3). To simplify, multiply numerator and denominator by sqrt(3): x = (9 * sqrt(3)) / (sqrt(3) * sqrt(3)) = (9 * sqrt(3)) / 3 = 3 * sqrt(3) cm.

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Step 5: Find the hypotenuse (side opposite 90-degree angle). The hypotenuse is 2 times the side opposite the 30-degree angle. So, hypotenuse = 2 * x = 2 * (3 * sqrt(3)) = 6 * sqrt(3) cm.

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Answer: The side opposite the 30-degree angle is 3 * sqrt(3) cm, and the hypotenuse is 6 * sqrt(3) cm.

Why It Matters

Understanding this special ratio is crucial in fields like Engineering and Architecture for designing structures and calculating forces efficiently. In Physics, it helps calculate projectile motion and light refraction. Even in Space Technology, it's used for satellite trajectory calculations.

Common Mistakes

MISTAKE: Confusing which side corresponds to which angle (e.g., putting '2' opposite 30 degrees). | CORRECTION: Always remember the smallest side (1x) is opposite the smallest angle (30 degrees), the medium side (sqrt(3)x) is opposite the medium angle (60 degrees), and the largest side (2x, hypotenuse) is opposite the largest angle (90 degrees).

MISTAKE: Forgetting to rationalize the denominator when solving for 'x' (e.g., leaving 9/sqrt(3) as the final answer). | CORRECTION: Always multiply the numerator and denominator by the square root to remove it from the denominator, making the answer simpler and easier to work with.

MISTAKE: Using the ratio 1:sqrt(2):1 for a 30-60-90 triangle. | CORRECTION: The 1:sqrt(2):1 ratio is for a 45-45-90 triangle. For a 30-60-90 triangle, the ratio is always 1:sqrt(3):2.

Practice Questions

Try It Yourself

QUESTION: In a 30-60-90 triangle, if the shortest side is 7 units, what is the length of the hypotenuse? | ANSWER: 14 units

QUESTION: The hypotenuse of a 30-60-90 triangle is 20 cm. What is the length of the side opposite the 60-degree angle? | ANSWER: 10 * sqrt(3) cm

QUESTION: A ladder leans against a wall, forming a 60-degree angle with the ground. If the base of the ladder is 4 meters from the wall, how long is the ladder, and how high up the wall does it reach? | ANSWER: Ladder length = 8 meters, Height up wall = 4 * sqrt(3) meters

MCQ

Quick Quiz

What is the ratio of the side lengths opposite the 30-degree, 60-degree, and 90-degree angles in a 30-60-90 triangle?

1 : 2 : sqrt(3)

1 : sqrt(3) : 2

2 : 1 : sqrt(3)

sqrt(3) : 1 : 2

The Correct Answer Is:

The correct ratio is 1 : sqrt(3) : 2, where 1 is opposite 30 degrees, sqrt(3) is opposite 60 degrees, and 2 is opposite 90 degrees. Option B correctly matches this sequence.

Real World Connection

In the Real World

From designing the slope of a roof in a new housing project to calculating the angle of a solar panel for maximum efficiency, the 30-60-90 triangle ratio is vital. Even ISRO scientists use these geometric principles for calculating rocket launch trajectories and satellite orbits.

Key Vocabulary

Key Terms

Right-angled triangle: A triangle with one angle measuring 90 degrees. | Hypotenuse: The longest side of a right-angled triangle, opposite the 90-degree angle. | Ratio: A comparison of two or more numbers. | Rationalize: The process of removing a radical from the denominator of a fraction.

What's Next

What to Learn Next

Now that you've mastered the 30-60-90 triangle, you're ready to explore the 45-45-90 triangle, another special right triangle. Understanding both will give you a strong foundation for trigonometry and advanced geometry problems!