The 45-45-90 triangle is one of the most important special right triangles in geometry. Because its sides follow a simple and predictable pattern, it is widely used in algebra, trigonometry, architecture, engineering, and standardized math tests.
Unlike ordinary right triangles, a 45-45-90 triangle has equal legs and fixed angle measures, making it much easier to solve without lengthy calculations.In this guide, you will learn:
- What a 45-45-90 triangle is
- The side ratio formula
- Why the ratio works
- Important properties and rules
- How to solve missing sides
- Area and perimeter formulas
- Real-life applications
- Solved examples and practice problems
What Is a 45-45-90 Triangle?
A 45-45-90 triangle is a special right triangle with angle measures:- $45^\circ$
- $45^\circ$
- $90^\circ$
Because the other two angles are equal, the triangle is also an isosceles triangle, meaning the two legs have equal lengths.
Side Ratio of a 45-45-90 Triangle
The sides of a 45-45-90 triangle always follow the ratio: $1:1:\sqrt{2}$This means:
- both legs are equal,
- the hypotenuse equals a leg multiplied by $\sqrt{2}$.
then the hypotenuse is: $x\sqrt{2}$
So the complete side relationship becomes: $x:x:x\sqrt{2}$
Why Does the Ratio Work?
The 45-45-90 triangle can be formed by cutting a square diagonally into two congruent triangles.Suppose the square has side length: $x$
The diagonal of the square becomes the hypotenuse of the triangle.
Using the Pythagorean Theorem: $a^2+b^2=c^2$
Since both legs are equal:
$x^2+x^2=c^2$
$2x^2=c^2$
$c=\sqrt{2x^2}$
$c=x\sqrt{2}$
Therefore, the side ratio is: $1:1:\sqrt{2}$
Properties of a 45-45-90 Triangle
1. It Is a Right Triangle
One angle always measures: $90^\circ$2. The Legs Are Equal
If one leg equals: $x$then the other leg is also: $x$
3. The Hypotenuse Contains $\sqrt{2}$
The hypotenuse always equals: $x\sqrt{2}$4. It Is an Isosceles Triangle
Since two sides are equal, the triangle is classified as an isosceles right triangle.5. The Angles Always Stay the Same
Every 45-45-90 triangle contains:- two $45^\circ$ angles,
- one $90^\circ$ angle.
How to Solve a 45-45-90 Triangle
Case 1: Given One Leg
If one leg equals: $x$then:
- other leg $=x$
- hypotenuse $=x\sqrt{2}$
Example
Suppose one leg is: $6$Then:
- Other leg: $6$
- Hypotenuse: $6\sqrt{2}$
Case 2: Given the Hypotenuse
If the hypotenuse equals: $x\sqrt{2}$then: $x=\frac{\text{hypotenuse}}{\sqrt{2}}$
Example
Suppose the hypotenuse is: $10\sqrt{2}$Then: Each leg: $\frac{10\sqrt{2}}{\sqrt{2}}=10$
Case 3: Using the Pythagorean Theorem
You can also solve the triangle using: $a^2+b^2=c^2$Since the legs are equal: $x^2+x^2=c^2$
Example
If each leg equals: $8$ then:- $8^2+8^2=c^2$
- $64+64=c^2$
- $128=c^2$
- $c=8\sqrt{2}$
Area of a 45-45-90 Triangle
The area formula for a triangle is: $\text{Area}=\frac{1}{2}\times \text{base}\times \text{height}$Since both legs are equal:
$\text{Area}=\frac{1}{2}\times x\times x$
$\text{Area}=\frac{x^2}{2}$
Perimeter of a 45-45-90 Triangle
Add all three sides: $x+x+x\sqrt{2}$Simplify: $2x+x\sqrt{2}$
So the perimeter formula becomes: $P=2x+x\sqrt{2}$
Solved Examples
Example 1
Find the hypotenuse if each leg measures: $9$Solution
Use the formula:$c=x\sqrt{2}$
$c=9\sqrt{2}$
So the hypotenuse is: $9\sqrt{2}$
Example 2
A 45-45-90 triangle has hypotenuse: $14\sqrt{2}$Find the legs.
Solution
Each leg equals: $\frac{14\sqrt{2}}{\sqrt{2}}$ $=14$So both legs measure: $14$
Example 3
Find the area of a 45-45-90 triangle with leg length: $12$Solution
Use:- $\text{Area}=\frac{x^2}{2}$
- $\text{Area}=\frac{12^2}{2}$
- $\text{Area}=\frac{144}{2}$
- $\text{Area}=72$
Common Mistakes Students Make
Confusing the Hypotenuse Formula
Remember: $x\sqrt{2}$not:$2x$Forgetting the Legs Are Equal
Both legs always have the same length.Mixing Up With the 30-60-90 Triangle
The ratios are different:- 45-45-90 triangle: $1:1:\sqrt{2}$
- 30-60-90 triangle: $1:\sqrt{3}:2$
Final Thoughts
The 45-45-90 triangle is one of the most useful geometric shapes in mathematics. Its equal legs and predictable side ratio make solving right-triangle problems much easier.By understanding the ratio: $1:1:\sqrt{2}$
students can quickly calculate unknown sides without complicated algebra.
Whether you are studying geometry, preparing for exams, solving engineering problems, or learning trigonometry, mastering the 45-45-90 triangle is an essential mathematical skill.
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