45 45 90 triangle formula proof properties and solved problems
A 45 45 90 triangle is a special case in geometry, commonly seen in school maths, competitive exams, and real-life applications. Its unique structure makes calculating side lengths and areas much easier, benefiting students in topics like trigonometry, algebra, and coordinate geometry. Understanding this triangle is essential for board exams, JEE, NEET, and beyond.
Understanding the 45 45 90 Triangle
A 45 45 90 triangle is a right triangle with two 45-degree angles and one 90-degree angle. This means the two legs (sides opposite 45° angles) are always equal. Since both legs are the same, it's also called an isosceles right triangle. The unique property of this triangle is that you can easily determine all sides and area if you know just one side.
Properties of a 45 45 90 Triangle
- Two equal angles: 45° each
- One right angle: 90°
- Two equal sides (legs)
- Hypotenuse is longer than either leg
- Sides are in the ratio 1 : 1 : √2 (leg : leg : hypotenuse)
- It’s half of a square, split along the diagonal
Formulae and Side Lengths in 45 45 90 Triangles
The special triangle ratio helps you quickly find missing sides:
| Given | Leg (a, b) | Hypotenuse (c) |
|---|---|---|
| Leg = x | x | x√2 |
| Hypotenuse = h | h/√2 | h |
Key formulas:
- Hypotenuse c = x√2 (if leg = x)
- Each Leg x = c / √2 (if hypotenuse = c)
- Area = (x × x) / 2 = x² / 2
- Perimeter = x + x + x√2 = x(2 + √2)
Step-by-Step Example: Finding Sides and Area
Let’s solve a problem.
Example: If one leg of a 45 45 90 triangle is 6 cm, find the hypotenuse and area.
- Find the hypotenuse:
Hypotenuse = leg × √2 = 6 × 1.414 = 8.49 cm (rounded to 2 decimals) - Find the area:
Area = (6 × 6) / 2 = 36 / 2 = 18 cm²
Practice Problems
- If the hypotenuse of a 45 45 90 triangle is 10 units, what are the lengths of the legs?
- Find the area of a 45 45 90 triangle with legs of 7 cm.
- The perimeter of a 45 45 90 triangle is 17.1 cm. Find the length of each leg (Round your answer to 2 decimal places).
- True/False: All isosceles right triangles are 45 45 90 triangles.
- If the area of a 45 45 90 triangle is 32 cm², what is the length of each leg?
Common Mistakes to Avoid
- Not applying the √2 factor to the hypotenuse (hypotenuse is not just double the leg).
- Confusing the side ratio with that of a 30 60 90 triangle.
- Forgetting to use the correct formula for area (should be leg × leg / 2).
- Rounding √2 too early; keep at least 2 decimals for accurate results.
Real-World Applications
45 45 90 triangles are everywhere – in carpentry, art, and architecture. Cutting a square diagonally creates two such triangles, useful for framing, tiling, and even bridge design. In maths, they’re essential for quick trigonometric calculations, and they appear often in coordinate geometry problems.
At Vedantu, we guide students through these applications and help make sense of geometry in exams and practical life. You can also explore related triangles like the isosceles triangle and apply these skills in trigonometry problems.
In summary, mastering the 45 45 90 triangle unlocks faster problem solving in maths exams and real-world design. This triangle’s consistent properties make side calculations, area finding, and geometry tasks much simpler. With regular practice and proper understanding, you’ll find geometry questions easier and more enjoyable.
FAQs on Understanding the 45 45 90 Triangle in Geometry
1. What is a 45 45 90 triangle?
A 45 45 90 triangle is a special right triangle with angles 45°, 45°, and 90° and two equal sides. Because the two acute angles are equal, it is also an isosceles right triangle.
- The two legs are equal in length.
- The 90° angle is between the equal sides.
- The hypotenuse is opposite the 90° angle.
2. What is the side ratio of a 45 45 90 triangle?
The side ratio of a 45 45 90 triangle is 1 : 1 : √2. This means:
- Both legs are equal (1 and 1).
- The hypotenuse is √2 times one leg.
3. What is the formula for the hypotenuse of a 45 45 90 triangle?
The hypotenuse of a 45 45 90 triangle is found using the formula hypotenuse = leg × √2. Since both legs are equal:
- If one leg = x
- Then hypotenuse = x√2
4. How do you find the legs of a 45 45 90 triangle if the hypotenuse is given?
To find each leg, divide the hypotenuse by √2. The formula is leg = hypotenuse ÷ √2.
- Given hypotenuse = h
- Each leg = h/√2
5. How do you solve a 45 45 90 triangle step by step?
To solve a 45 45 90 triangle, use its fixed side ratio 1 : 1 : √2.
- Step 1: Identify the given side (leg or hypotenuse).
- Step 2: Use the ratio to find missing sides.
- Step 3: Apply hypotenuse = leg × √2 or leg = hypotenuse ÷ √2.
6. Why is a 45 45 90 triangle called an isosceles right triangle?
A 45 45 90 triangle is called an isosceles right triangle because it has two equal sides and one right angle. In this triangle:
- The two equal sides form the 90° angle.
- The opposite angles are both 45°.
7. How is the Pythagorean theorem used in a 45 45 90 triangle?
The Pythagorean theorem shows why the hypotenuse equals leg × √2 in a 45 45 90 triangle. If each leg is x:
- x² + x² = hypotenuse²
- 2x² = hypotenuse²
- hypotenuse = x√2
8. What is the area of a 45 45 90 triangle?
The area of a 45 45 90 triangle is (1/2) × leg × leg because the legs are perpendicular. Since both legs are equal (x):
- Area = (1/2)x²
9. What is the difference between a 45 45 90 triangle and a 30 60 90 triangle?
The main difference is their angle measures and side ratios.
- 45 45 90 triangle: angles 45°, 45°, 90° and ratio 1 : 1 : √2.
- 30 60 90 triangle: angles 30°, 60°, 90° and ratio 1 : √3 : 2.
10. Where are 45 45 90 triangles used in real life?
A 45 45 90 triangle is commonly used in geometry, construction, architecture, and coordinate geometry.
- Designing square frames and diagonal supports.
- Finding distances using diagonals of squares.
- Solving problems involving slopes of 1 (rise equals run).
