Special Right Triangles Worksheet With Answers For Students
Table of Contents :
Understanding special right triangles is crucial in geometry and trigonometry. Special right triangles include the 45-45-90 triangle and the 30-60-90 triangle, both of which have unique properties that simplify calculations. In this article, we will explore the key characteristics of these triangles, provide a worksheet for practice, and offer answers to enhance learning. Let's dive in! 📐
Special Right Triangles
45-45-90 Triangle
The 45-45-90 triangle is an isosceles right triangle, meaning that the two legs are equal in length. The relationships between the sides are as follows:
- If the length of each leg is ( x ), then the length of the hypotenuse ( h ) can be calculated using the formula: [ h = x \sqrt{2} ]
30-60-90 Triangle
The 30-60-90 triangle has sides in a specific ratio that makes it easier to calculate the lengths of each side. The relationships between the sides are:
- If the length of the shortest leg (opposite the 30-degree angle) is ( x ), then:
- The length of the longer leg (opposite the 60-degree angle) is: [ \text{Longer leg} = x \sqrt{3} ]
- The length of the hypotenuse ( h ) is: [ h = 2x ]
Key Formulas for Special Right Triangles
Below is a summary of key formulas to remember for both types of special right triangles:
<table> <tr> <th>Triangle Type</th> <th>Legs</th> <th>Hypotenuse</th> </tr> <tr> <td>45-45-90</td> <td>Both legs = x</td> <td>Hypotenuse = ( x \sqrt{2} )</td> </tr> <tr> <td>30-60-90</td> <td>Short leg = x</td> <td>Long leg = ( x \sqrt{3} )</td> <td>Hypotenuse = ( 2x )</td> </tr> </table>
Worksheet: Special Right Triangles
Now that we have a solid understanding of special right triangles, let's practice! Below is a worksheet designed to test your knowledge of these triangles.
Instructions
- For each triangle below, determine the length of the missing sides using the properties of special right triangles.
- Show your work for each calculation.
Problems
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In a 45-45-90 triangle, if each leg is 8 cm, what is the length of the hypotenuse?
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In a 30-60-90 triangle, if the shortest leg measures 5 cm, calculate the lengths of the longer leg and the hypotenuse.
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If the hypotenuse of a 45-45-90 triangle is ( 10\sqrt{2} ) cm, what is the length of each leg?
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For a 30-60-90 triangle, if the longer leg is ( 8\sqrt{3} ) cm, what are the lengths of the shorter leg and the hypotenuse?
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In a 45-45-90 triangle, if the hypotenuse measures 14 cm, find the length of each leg.
Notes:
Remember to use the relationships between the sides of the triangles for your calculations!
Answers to the Worksheet
Here are the solutions to the problems listed in the worksheet. Review your work and see how you did! 📝
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Hypotenuse Calculation:
- ( h = x\sqrt{2} = 8\sqrt{2} ) cm
- Answer: Hypotenuse = ( 8\sqrt{2} ) cm
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Leg Lengths Calculation:
- Longer leg = ( 5\sqrt{3} ) cm
- Hypotenuse = ( 10 ) cm
- Answer: Longer leg = ( 5\sqrt{3} ) cm, Hypotenuse = ( 10 ) cm
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Leg Calculation:
- Each leg ( x = \frac{10\sqrt{2}}{\sqrt{2}} = 10 ) cm
- Answer: Each leg = 10 cm
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Legs Calculation:
- Short leg ( x = \frac{8\sqrt{3}}{\sqrt{3}} = 8 ) cm
- Hypotenuse = ( 2 \times 8 = 16 ) cm
- Answer: Short leg = 8 cm, Hypotenuse = 16 cm
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Leg Calculation:
- Each leg ( x = \frac{14}{\sqrt{2}} = 7\sqrt{2} ) cm
- Answer: Each leg = ( 7\sqrt{2} ) cm
Conclusion
Understanding special right triangles is essential for solving many geometric problems. By practicing with the worksheet, you can strengthen your skills and enhance your confidence in tackling right triangle questions. Remember to use the unique properties and ratios associated with these triangles to simplify your calculations! Happy studying! 📚✨