The Hypotenuse Of A $45^{\circ}-45^{\circ}-90^{\circ}$ Triangle Measures $10 \sqrt{5}$ In.What Is The Length Of One Leg Of The Triangle?A. \$5 \sqrt{5}$[/tex\]B. $5 \sqrt{10}$C. $10 \sqrt{5}$D.

by ADMIN 203 views

Introduction

In the realm of geometry, triangles are fundamental shapes that have 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 world of $45{\circ}-45{\circ}-90^{\circ}$ triangles and explore the relationship between the sides, particularly the hypotenuse and the legs.

The 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 equal acute angles, each measuring $45^{\circ}$. The third angle, the right angle, measures $90^{\circ}$. This type of triangle has several distinct properties that set it apart from other types of triangles.

One of the key properties of a $45{\circ}-45{\circ}-90^{\circ}$ triangle is that the two legs are equal in length. This means that if one leg measures $x$ units, the other leg also measures $x$ units. The hypotenuse, which is the side opposite the right angle, is always $\sqrt{2}$ times the length of each leg.

The Relationship Between the Hypotenuse and the Legs

Given that the hypotenuse measures $10 \sqrt{5}$ in, we can use the properties of a $45{\circ}-45{\circ}-90^{\circ}$ triangle to find the length of one leg. Since the hypotenuse is $\sqrt{2}$ times the length of each leg, we can set up the following equation:

2×leg=105\sqrt{2} \times \text{leg} = 10 \sqrt{5}

To solve for the length of the leg, we can divide both sides of the equation by $\sqrt{2}$:

leg=1052\text{leg} = \frac{10 \sqrt{5}}{\sqrt{2}}

Simplifying the Expression

To simplify the expression, we can rationalize the denominator by multiplying both the numerator and the denominator by $\sqrt{2}$:

leg=105×22×2\text{leg} = \frac{10 \sqrt{5} \times \sqrt{2}}{\sqrt{2} \times \sqrt{2}}

leg=10102\text{leg} = \frac{10 \sqrt{10}}{2}

leg=510\text{leg} = 5 \sqrt{10}

Conclusion

In conclusion, the length of one leg of the $45{\circ}-45{\circ}-90^{\circ}$ triangle is $5 \sqrt{10}$. This is a direct result of the properties of a $45{\circ}-45{\circ}-90^{\circ}$ triangle and the relationship between the hypotenuse and the legs.

Discussion

The relationship between the hypotenuse and the legs of a $45{\circ}-45{\circ}-90^{\circ}$ triangle is a fundamental concept in geometry. Understanding this relationship is crucial for solving problems involving right-angled triangles. In this article, we have demonstrated how to use the properties of a $45{\circ}-45{\circ}-90^{\circ}$ triangle to find the length of one leg given the length of the hypotenuse.

Final Answer

The final answer is: 510\boxed{5 \sqrt{10}}

Introduction

In our previous article, we explored the properties of a $45{\circ}-45{\circ}-90^{\circ}$ triangle and how to find 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.

Q1: What is the relationship between the hypotenuse and the legs of a $45{\circ}-45{\circ}-90^{\circ}$ triangle?

A1: The hypotenuse of a $45{\circ}-45{\circ}-90^{\circ}$ triangle is always $\sqrt{2}$ times the length of each leg. This means that if one leg measures $x$ units, the hypotenuse measures $x\sqrt{2}$ units.

Q2: How do I find the length of one leg of a $45{\circ}-45{\circ}-90^{\circ}$ triangle given the length of the hypotenuse?

A2: To find the length of one leg of a $45{\circ}-45{\circ}-90^{\circ}$ triangle given the length of the hypotenuse, you can use the following formula:

leg=hypotenuse2\text{leg} = \frac{\text{hypotenuse}}{\sqrt{2}}

Q3: What is the length of the hypotenuse of a $45{\circ}-45{\circ}-90^{\circ}$ triangle if one leg measures $5$ units?

A3: To find the length of the hypotenuse of a $45{\circ}-45{\circ}-90^{\circ}$ triangle if one leg measures $5$ units, you can use the following formula:

hypotenuse=leg×2\text{hypotenuse} = \text{leg} \times \sqrt{2}

hypotenuse=5×2\text{hypotenuse} = 5 \times \sqrt{2}

hypotenuse=52\text{hypotenuse} = 5\sqrt{2}

Q4: How do I find the length of the hypotenuse of a $45{\circ}-45{\circ}-90^{\circ}$ triangle if one leg measures $10$ units?

A4: To find the length of the hypotenuse of a $45{\circ}-45{\circ}-90^{\circ}$ triangle if one leg measures $10$ units, you can use the following formula:

hypotenuse=leg×2\text{hypotenuse} = \text{leg} \times \sqrt{2}

hypotenuse=10×2\text{hypotenuse} = 10 \times \sqrt{2}

hypotenuse=102\text{hypotenuse} = 10\sqrt{2}

Q5: What is the relationship between the hypotenuse and the legs of a $45{\circ}-45{\circ}-90^{\circ}$ triangle in terms of the ratio of the sides?

A5: The ratio of the sides of a $45{\circ}-45\circ}-90^{\circ}$ triangle is $11:\sqrt{2$. This means that the ratio of the length of one leg to the length of the hypotenuse is $1:\sqrt{2}$.

Conclusion

In conclusion, the relationship between the hypotenuse and the legs of a $45{\circ}-45{\circ}-90^{\circ}$ triangle is a fundamental concept in geometry. Understanding this relationship is crucial for solving problems involving right-angled triangles. We hope that this Q&A article has provided you with a better understanding of the properties of a $45{\circ}-45{\circ}-90^{\circ}$ triangle and how to find the length of one leg given the length of the hypotenuse.

Final Answer

The final answer is: 510\boxed{5 \sqrt{10}}