45 45 90 Triangle Worksheet and Answer Key
If you’re teaching geometry to your middle school or high school student this year, chances are you’ve encountered the 45-45-90 special right triangles.
This flavor of isosceles triangle problem may seem overwhelming to solve (and to teach!), but with a few tips, you’ll be finding those missing side lengths like a pro in no time!
To help you out, I’ve created not one but two worksheets to help you teach and review calculating the length of the sides of a 45-45-90 triangle.
The first worksheet has simple problems to help you build confidence, while the second one offers more complicated challenges for those ready to level up. And of course, each worksheet comes with an answer key!
*****Psst! I know you’re in a mega hurry to get to printing and teaching, so rest assured that your worksheet packet download link is at the bottom of this blog post. No email is required 🙂
Check Out Your Sides Of a 45-45-90 Triangle Worksheet Packet!
1. Simple Version of Worksheet
2. Advanced Version of Worksheet
****Again, your download link is at the bottom of this post, but don’t miss all the great teaching tips below!
What Exactly Is A 45-45-90 Triangle?
Let’s start with the basics. A 45-45-90 triangle is an isosceles right triangle.
This means it’s a right triangle with two equal sides and two 45-degree angles. It also happens to be half of a square cut along its diagonal.
The two sides forming the 90-degree angle are always the same length, making it a special triangle with some unique properties.
Why Are 45-45-90 Triangles Special?
The beauty of 45-45-90 triangles lies in their predictable side relationships. Here’s why they’re so special:
- The two legs (the sides that form the right angle) are of equal length.
- The hypotenuse (the side opposite the 90-degree angle) is the length of a leg multiplied by the square root of 2 (√2).
So, if you know the length of a leg (let’s call it x), you can instantly know the length of the other leg (also x) and the hypotenuse length (x√2). This makes solving for the sides of the triangle straightforward.
How To Solve The Problems On The Worksheet
To get comfortable with these triangle rules, let’s walk through some step-by-step instructions for a couple of example problems
Example Problem 1: Finding the Missing Side Lengths
Problem:
You have a 45-45-90 triangle where one leg measures 7. Find the length of the other leg and the hypotenuse. Leave your answer in simplest radical form.
Solution:
- Identify What You Know:
The length of the leg measures 7. Since it’s an isosceles right triangle, the other leg is also 7. - Calculate the Hypotenuse (longest side):
The hypotenuse length in a 45-45-90 triangle is the leg length multiplied by √2.
Hypotenuse = 7 × √2 = 7√2. - Write the Final Answer:
- The other leg = 7
- The hypotenuse = 7√2
You’ve found the missing side lengths just like that!
Example Problem 2: Solving When Given the Hypotenuse
Problem:
Find the missing side lengths in a 45-45-90 triangle where the hypotenuse is 10√2.
Solution:
- Identify What You Know:
You know the hypotenuse length is 10√2. - Set Up the Equation:
In a 45-45-90 triangle, the hypotenuse is always x√2 (where x is the length of each leg). Set the hypotenuse equal to 10√2:
x√2 = 10√2. - Solve for x:
Divide both sides by √2:
x = 10. So, each leg is 10. - Write the Final Answer:
- The two legs = 10
- The hypotenuse = 10√2
And just like that, you’ve solved for the missing side lengths again!
Strategy Recap: Key Points to Remember
Now that you’ve seen a few examples, here’s a quick recap of strategies to tackle any trig questions involving 45-45-90 triangles:
- Recognize the triangle type: Understand you’re dealing with an isosceles right triangle with congruent sides.
- Apply the side length relationships: Remember, for a 45-45-90 triangle:
- Leg = x
- Hypotenuse = x√2
This works because the triangle is literally the diagonal of a square cut in half.
- Utilize the Pythagorean Theorem formula: If you’re unsure, use the Pythagorean theorem formula (a² + b² = c²) to check your work.
- Use a triangle calculator: Sometimes, a little tech goes a long way. An online triangle calculator can help verify your answers quickly.
Going Beyond: Tips for High School Students
If you’re looking for additional ways to strengthen your understanding, consider these approaches:
- Use external resources: Videos, online tutorials, and math websites can provide more examples and step-by-step explanations.
- Practice with trig functions: Get familiar with trigonometric functions like sine, cosine, and tangent, which often involve the exact values found in special triangles like the 45-45-90 triangle.
- Explore related concepts: Understanding 30-60-90 triangles and how they differ from 45-45-90 triangles will broaden your understanding of right angle triangles.
- Test your knowledge with a self-checking Google sheet: Create or find a self-checking Google sheet to practice problems and immediately know if your answers are correct.
- Utilize Google Classroom: If your class uses Google Classroom, check for any assignments or resources your teacher may have provided.
Applying Your Skills In The Real World
Understanding 45-45-90 triangles isn’t just about passing tests.
These concepts help in real-world applications, like finding the length of the diagonal of a square, calculating the perimeter of a triangle, or even solving complex trig questions.
Click Here To Download Your 45-45-90 Worksheet and Answer Key Packet!
****Click the above image link/text link to get your triangle worksheets.
By now, you should feel more confident about tackling 45-45-90 triangles and their missing side lengths.
Remember, practice makes perfect, so use the worksheets—one for starting out and another for more advanced problems—to build your skills step-by-step.
And don’t forget to check your answers with the key to see where you stand!
Happy studying!
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