Geometry · Step-by-step guide
Special Right Triangles
Use 45-45-90 and 30-60-90 triangle ratios to find exact side lengths quickly.
Start with the central idea
A 45-45-90 triangle has side ratio . A 30-60-90 triangle has ratio , ordered as short leg, long leg, hypotenuse.
Ratio: x : x : x√2
Ratio: x : x√3 : 2x
How it works
Build the method from meaning before memorizing the moves.
These are the decisions that make the procedure reliable. Read them once, then look for each idea in the worked examples below.
Match angle to side
The smallest side lies opposite 30°, equal legs lie opposite equal 45° angles, and the hypotenuse lies opposite 90°.
Keep exact radicals
Special-triangle problems usually expect simplified radical form rather than early decimal approximations.
Scale the whole ratio
Once one side fixes x, multiply every ratio entry by that same value.
See the structure
A picture makes the relationships easier to remember.
Worked examples
Follow the reason for each line, then try to reproduce it without looking.
A leg is 7.
- Both legs are equal.
- The hypotenuse is leg times .
- Keep the exact radical.
The short leg is 5.
- Set .
- The long leg is .
- The hypotenuse is .
The hypotenuse is 18.
- , so .
- The short leg is 9.
- The long leg is .
Common mistakes—and how to avoid them
Accuracy improves fastest when you know what to check.
Putting √3 on the short leg.
Using the ratios without first identifying the angles.
Turning exact radicals into rounded decimals too early.
Turn the explanation into a skill
Reading creates recognition. Independent practice creates recall.
Triangle Properties
Reinforce triangle structure, angle relationships, and side reasoning.
Preview the worksheetA strong practice loop: solve one example with the guide open, solve a similar problem without it, explain the method aloud, and return the next day for a short mixed review.
Practice problems with worked solutions
Solve each problem on paper, use the hint only if needed, then compare every step.
Use the 45–45–90 triangle diagram to find hypotenuse c.
Show hint
Multiply a leg by .
Show worked solution
- The ratio is .
- With scale factor 7, the hypotenuse is .
Answer:
Use the 45–45–90 triangle diagram to find each leg x.
Show hint
Divide the hypotenuse by .
Show worked solution
- .
- Both legs are equal.
Answer:
Use the 30–60–90 triangle diagram to find the long leg and hypotenuse.
Show hint
Use in short-leg, long-leg, hypotenuse order.
Show worked solution
- Multiply the short leg 4 by for the long leg.
- Double the short leg for the hypotenuse.
Answer: Long leg ; hypotenuse 8.
Use the 30–60–90 triangle diagram to find both legs.
Show hint
The short leg is half the hypotenuse.
Show worked solution
- Short leg .
- Long leg .
Answer: Short leg 9; long leg .
Use the square and its diagonal to find side length s.
Show hint
The diagonal creates a 45–45–90 triangle.
Show worked solution
- The diagonal is the hypotenuse, so .
- .
Answer:
Frequently asked questions
Quick answers before you move on.
What should I remember first?
A 45-45-90 triangle has side ratio . A 30-60-90 triangle has ratio , ordered as short leg, long leg, hypotenuse.
How do I know whether I understand this topic?
You should be able to name the method, explain why each step is valid, complete a new example without copying, and check whether your answer is reasonable.
What should I do if I keep making the same mistake?
Write the error as a specific checkpoint—for example, “I will identify the hypotenuse before substituting.” Then use that checkpoint on three short problems rather than repeating a full page without feedback.