The Hypotenuse Of A $45^{\circ}-45^{\circ}-90^{\circ}$ Triangle Measures $22 \sqrt{2}$ Units.What Is The Length Of One Leg Of The Triangle?A. 11 Units B. \$11 \sqrt{2}$[/tex\] Units C. 22 Units D. $22

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Introduction

In the realm of geometry, triangles are a fundamental concept that has been studied extensively. Among the various types of triangles, the $45{\circ}-45{\circ}-90^{\circ}$ triangle is a special case that has unique properties. In this article, we will delve into the relationship between the legs and the hypotenuse of a $45{\circ}-45{\circ}-90^{\circ}$ triangle, with a focus on understanding how to calculate the length of one leg given the length of the hypotenuse.

Properties of a $45{\circ}-45{\circ}-90^{\circ}$ Triangle

A $45{\circ}-45{\circ}-90^{\circ}$ triangle is a right-angled triangle with two acute angles, each measuring $45^{\circ}$. The side opposite the right angle is called the hypotenuse, and the other two sides are called the legs. In a $45{\circ}-45{\circ}-90^{\circ}$ triangle, the two legs are equal in length, and the hypotenuse is $\sqrt{2}$ times the length of each leg.

Relationship Between Legs and Hypotenuse

The relationship between the legs and the hypotenuse of a $45{\circ}-45{\circ}-90^{\circ}$ triangle can be expressed mathematically as:

Hypotenuse=2×Leg\text{Hypotenuse} = \sqrt{2} \times \text{Leg}

This means that if we know the length of the hypotenuse, we can calculate the length of one leg by dividing the length of the hypotenuse by $\sqrt{2}$.

Calculating the Length of One Leg

Given that the hypotenuse of a $45{\circ}-45{\circ}-90^{\circ}$ triangle measures $22 \sqrt{2}$ units, we can use the relationship between the legs and the hypotenuse to calculate the length of one leg.

Leg=Hypotenuse2\text{Leg} = \frac{\text{Hypotenuse}}{\sqrt{2}}

Substituting the given value of the hypotenuse, we get:

Leg=2222\text{Leg} = \frac{22 \sqrt{2}}{\sqrt{2}}

Simplifying the expression, we get:

Leg=22\text{Leg} = 22

Therefore, the length of one leg of the triangle is $22$ units.

Conclusion

In conclusion, the relationship between the legs and the hypotenuse of a $45{\circ}-45{\circ}-90^{\circ}$ triangle is a fundamental concept in geometry. By understanding this relationship, we can calculate the length of one leg given the length of the hypotenuse. In this article, we have seen how to use this relationship to calculate the length of one leg of a $45{\circ}-45{\circ}-90^{\circ}$ triangle, with a hypotenuse measuring $22 \sqrt{2}$ units.

Frequently Asked Questions

  • What is the relationship between the legs and the hypotenuse of a $45{\circ}-45{\circ}-90^{\circ}$ triangle?
    • The hypotenuse is $\sqrt{2}$ times the length of each leg.
  • How can we calculate the length of one leg given the length of the hypotenuse?
    • We can use the formula: $\text{Leg} = \frac{\text{Hypotenuse}}{\sqrt{2}}$
  • What is the length of one leg of a $45{\circ}-45{\circ}-90^{\circ}$ triangle with a hypotenuse measuring $22 \sqrt{2}$ units?
    • The length of one leg is $22$ units.

References

  • [1] "Geometry" by Michael Artin
  • [2] "Trigonometry" by I.M. Gelfand
  • [3] "Mathematics for the Nonmathematician" by Morris Kline

Further Reading

  • [1] "The Pythagorean Theorem" by Euclid
  • [2] "Trigonometry: A Unit Circle Approach" by Michael Sullivan
  • [3] "Geometry: Seeing, Doing, Understanding" by Harold R. Jacobs

Introduction

In our previous article, we explored the relationship between the legs and the hypotenuse of a $45{\circ}-45{\circ}-90^{\circ}$ triangle. We also calculated the length of one leg given the length of the hypotenuse. In this article, we will answer some frequently asked questions related to the hypotenuse of a $45{\circ}-45{\circ}-90^{\circ}$ triangle.

Q&A

Q: What is the relationship between the legs and the hypotenuse of a $45{\circ}-45{\circ}-90^{\circ}$ triangle?

A: The hypotenuse is $\sqrt{2}$ times the length of each leg.

Q: How can we calculate the length of one leg given the length of the hypotenuse?

A: We can use the formula: $\text{Leg} = \frac{\text{Hypotenuse}}{\sqrt{2}}$

Q: What is the length of one leg of a $45{\circ}-45{\circ}-90^{\circ}$ triangle with a hypotenuse measuring $22 \sqrt{2}$ units?

A: The length of one leg is $22$ units.

Q: Can we use the same formula to calculate the length of the hypotenuse given the length of one leg?

A: Yes, we can use the formula: $\text{Hypotenuse} = \sqrt{2} \times \text{Leg}$

Q: What is the relationship between the two legs of a $45{\circ}-45{\circ}-90^{\circ}$ triangle?

A: The two legs are equal in length.

Q: Can we use the Pythagorean theorem to calculate the length of the hypotenuse of a $45{\circ}-45{\circ}-90^{\circ}$ triangle?

A: No, we cannot use the Pythagorean theorem to calculate the length of the hypotenuse of a $45{\circ}-45{\circ}-90^{\circ}$ triangle. The Pythagorean theorem is used to calculate the length of the hypotenuse of a right triangle, but it is not necessary in this case since we already know the relationship between the legs and the hypotenuse.

Q: Can we use the formula to calculate the length of one leg of a $45{\circ}-45{\circ}-90^{\circ}$ triangle with a hypotenuse measuring $22$ units?

A: Yes, we can use the formula: $\text{Leg} = \frac{\text{Hypotenuse}}{\sqrt{2}}$

Q: What is the length of the hypotenuse of a $45{\circ}-45{\circ}-90^{\circ}$ triangle with a leg measuring $11$ units?

A: The length of the hypotenuse is $22$ units.

Conclusion

In conclusion, the relationship between the legs and the hypotenuse of a $45{\circ}-45{\circ}-90^{\circ}$ triangle is a fundamental concept in geometry. By understanding this relationship, we can calculate the length of one leg given the length of the hypotenuse. We have also answered some frequently asked questions related to the hypotenuse of a $45{\circ}-45{\circ}-90^{\circ}$ triangle.

Frequently Asked Questions

  • What is the relationship between the legs and the hypotenuse of a $45{\circ}-45{\circ}-90^{\circ}$ triangle?
  • How can we calculate the length of one leg given the length of the hypotenuse?
  • What is the length of one leg of a $45{\circ}-45{\circ}-90^{\circ}$ triangle with a hypotenuse measuring $22 \sqrt{2}$ units?
  • Can we use the same formula to calculate the length of the hypotenuse given the length of one leg?
  • What is the relationship between the two legs of a $45{\circ}-45{\circ}-90^{\circ}$ triangle?
  • Can we use the Pythagorean theorem to calculate the length of the hypotenuse of a $45{\circ}-45{\circ}-90^{\circ}$ triangle?
  • Can we use the formula to calculate the length of one leg of a $45{\circ}-45{\circ}-90^{\circ}$ triangle with a hypotenuse measuring $22$ units?
  • What is the length of the hypotenuse of a $45{\circ}-45{\circ}-90^{\circ}$ triangle with a leg measuring $11$ units?

References

  • [1] "Geometry" by Michael Artin
  • [2] "Trigonometry" by I.M. Gelfand
  • [3] "Mathematics for the Nonmathematician" by Morris Kline

Further Reading

  • [1] "The Pythagorean Theorem" by Euclid
  • [2] "Trigonometry: A Unit Circle Approach" by Michael Sullivan
  • [3] "Geometry: Seeing, Doing, Understanding" by Harold R. Jacobs