What is the Ratio of Sides in a 45-45-90 Triangle?

S6-SA2-0139

Grade Level:

Class 10

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

Definition

What is it?

A 45-45-90 triangle is a special right-angled triangle where two angles are 45 degrees and one is 90 degrees. Because two angles are equal, it's also an isosceles triangle, meaning the two sides opposite the 45-degree angles are equal in length. The ratio of its sides is 1 : 1 : sqrt(2).

Simple Example

Quick Example

Imagine a square-shaped 'rangoli' design. If you draw a diagonal line across this square, you divide it into two identical 45-45-90 triangles. If the sides of the square are 10 cm each, then the two shorter sides of the triangle are 10 cm, and the diagonal (hypotenuse) will be 10 * sqrt(2) cm.

Worked Example

Step-by-Step

Let's find the sides of a 45-45-90 triangle if one of the equal sides is 5 units long.
STEP 1: Identify the given information. We have a 45-45-90 triangle. One of the equal sides (opposite a 45-degree angle) is 5 units.
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STEP 2: Recall the side ratio for a 45-45-90 triangle: side : side : hypotenuse = 1 : 1 : sqrt(2).
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STEP 3: Since the two sides opposite the 45-degree angles are equal, the other equal side will also be 5 units.
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STEP 4: To find the hypotenuse, we use the ratio. If '1' in the ratio corresponds to 5 units, then 'sqrt(2)' corresponds to 5 * sqrt(2) units.
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STEP 5: Alternatively, use the Pythagorean theorem: a^2 + b^2 = c^2. Here, a=5, b=5. So, 5^2 + 5^2 = c^2.
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STEP 6: 25 + 25 = c^2, which means 50 = c^2.
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STEP 7: c = sqrt(50) = sqrt(25 * 2) = 5 * sqrt(2) units.
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ANSWER: The sides of the triangle are 5 units, 5 units, and 5 * sqrt(2) units.

Why It Matters

Understanding special triangles like 45-45-90 is crucial in fields like Engineering and Physics for quickly solving problems related to forces and structures. In Space Technology, calculating distances and angles for satellite orbits often uses these basic geometric principles. Even in AI/ML, understanding spatial relationships is key for computer vision applications.

Common Mistakes

MISTAKE: Assuming the hypotenuse is just double the side length (e.g., if side is 5, hypotenuse is 10). | CORRECTION: The hypotenuse is actually 'side * sqrt(2)', not 'side * 2'. Remember sqrt(2) is approximately 1.414.

MISTAKE: Confusing the 45-45-90 ratio with the 30-60-90 triangle ratio. | CORRECTION: Each special right triangle has a unique side ratio. For 45-45-90, it's 1:1:sqrt(2). For 30-60-90, it's 1:sqrt(3):2.

MISTAKE: Not recognizing that the two shorter sides are equal in a 45-45-90 triangle. | CORRECTION: Since the angles opposite these sides are both 45 degrees, the sides must be equal in length. This is a key property of isosceles triangles.

Practice Questions

Try It Yourself

QUESTION: If the hypotenuse of a 45-45-90 triangle is 8 cm, what is the length of each of the equal sides? | ANSWER: 4 * sqrt(2) cm or approximately 5.66 cm

QUESTION: A square park has a diagonal path of 20 meters. If you consider one of the 45-45-90 triangles formed by the path, what is the perimeter of this triangle? | ANSWER: 20 + 10 * sqrt(2) + 10 * sqrt(2) = 20 + 20 * sqrt(2) meters or approximately 48.28 meters

QUESTION: A carpenter needs to cut a right-angled triangular piece of wood. If the two equal sides are 7 cm each, what will be the length of the longest side, and what are the angles of the triangle? | ANSWER: Longest side = 7 * sqrt(2) cm; Angles are 45 degrees, 45 degrees, and 90 degrees.

MCQ

Quick Quiz

What is the ratio of the two shorter sides to the hypotenuse in a 45-45-90 triangle?

1 : 2 : sqrt(3)

1 : 1 : sqrt(2)

1 : sqrt(3) : 2

1 : 1 : 2

The Correct Answer Is:

The correct ratio for a 45-45-90 triangle is 1:1:sqrt(2), where the two '1's represent the equal sides and 'sqrt(2)' represents the hypotenuse. Options A and C are ratios for other special triangles, and D is incorrect.

Real World Connection

In the Real World

When architects design buildings or bridges, they often use 45-degree angles for stability and aesthetics. For example, the support beams in a roof or a bridge might form 45-45-90 triangles. Knowing this ratio helps engineers quickly calculate the required length of materials without complex calculations, making construction faster and more efficient.

Key Vocabulary

Key Terms

RIGHT-ANGLED TRIANGLE: A triangle with one angle exactly 90 degrees. | ISOSCELES TRIANGLE: A triangle with two sides of equal length and two equal angles. | HYPOTENUSE: The longest side of a right-angled triangle, opposite the 90-degree angle. | RATIO: A comparison of two or more numbers, showing their relative sizes.

What's Next

What to Learn Next

Now that you understand 45-45-90 triangles, you should explore the 'What is the Ratio of Sides in a 30-60-90 Triangle?' concept. It's another special right triangle with its own unique side ratio, and mastering both will give you a strong foundation for trigonometry!