If F G = 2 FG = 2 FG = 2 Units, F I = 7 FI = 7 F I = 7 Units, And H I = 1 HI = 1 H I = 1 Unit, What Is G H GH G H ?A. 3 Units B. 4 Units C. 5 Units D. 6 Units

Introduction

In geometry, problems involving triangles and their side lengths are common. These problems often require the application of various geometric concepts, such as the Pythagorean theorem, to find the unknown side lengths. In this article, we will solve a problem involving a triangle with given side lengths and find the length of a specific side.

Problem Statement

Given a triangle with side lengths $FG = 2$ units, $FI = 7$ units, and $HI = 1$ unit, we need to find the length of side $GH$. This problem can be solved using the Pythagorean theorem, which states that in a right-angled triangle, the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the lengths of the other two sides.

Applying the Pythagorean Theorem

To find the length of side $GH$, we can use the Pythagorean theorem. However, we need to find the length of side $FH$ first. We can do this by using the Pythagorean theorem on triangle $FIH$. The length of side $FH$ can be found using the formula:

$$FH = \sqrt{FI^2 - HI^2}$$

Substituting the given values, we get:

$$FH = \sqrt{7^2 - 1^2}$$

$$FH = \sqrt{49 - 1}$$

$$FH = \sqrt{48}$$

$$FH = 4\sqrt{3}$$

Finding the Length of GH

Now that we have the length of side $FH$, we can use the Pythagorean theorem on triangle $FHG$ to find the length of side $GH$. The length of side $GH$ can be found using the formula:

$$GH = \sqrt{FG^2 + FH^2}$$

Substituting the given values, we get:

$$GH = \sqrt{2^2 + (4\sqrt{3})^2}$$

$$GH = \sqrt{4 + 48}$$

$$GH = \sqrt{52}$$

$$GH = 2\sqrt{13}$$

Conclusion

In this article, we solved a geometric problem involving a triangle with given side lengths. We used the Pythagorean theorem to find the length of side $GH$. The length of side $GH$ is $2\sqrt{13}$ units.

Answer

The correct answer is not listed among the options A, B, C, or D. However, we can approximate the value of $2\sqrt{13}$ to be around 5.66 units. This is closest to option C, which is 5 units.

Final Answer

The final answer is $\boxed{5}$ units.

Introduction

In our previous article, we solved a geometric problem involving a triangle with given side lengths and found the length of a specific side. In this article, we will provide a Q&A guide to help you understand the concepts and techniques used to solve geometric problems.

Q: What is the Pythagorean Theorem?

A: The Pythagorean theorem is a fundamental concept in geometry that states that in a right-angled triangle, the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the lengths of the other two sides. The formula is:

$$a^2 + b^2 = c^2$$

where $a$ and $b$ are the lengths of the two sides, and $c$ is the length of the hypotenuse.

Q: How do I apply the Pythagorean Theorem?

A: To apply the Pythagorean theorem, you need to identify the right-angled triangle and the lengths of the two sides. Then, you can use the formula to find the length of the hypotenuse. For example, if you have a triangle with side lengths $a = 3$ units and $b = 4$ units, you can use the formula to find the length of the hypotenuse $c$:

$$c^2 = a^2 + b^2$$

$$c^2 = 3^2 + 4^2$$

$$c^2 = 9 + 16$$

$$c^2 = 25$$

$$c = \sqrt{25}$$

$$c = 5$$

Q: What is the difference between a right-angled triangle and an oblique triangle?

A: A right-angled triangle is a triangle with one right angle (90 degrees). An oblique triangle is a triangle with no right angles. In a right-angled triangle, the Pythagorean theorem can be used to find the length of the hypotenuse. In an oblique triangle, the Pythagorean theorem cannot be used, and other techniques such as the law of cosines or the law of sines may be used.

Q: How do I find the length of a side in a triangle?

A: To find the length of a side in a triangle, you need to use the appropriate formula or technique. For a right-angled triangle, you can use the Pythagorean theorem. For an oblique triangle, you can use the law of cosines or the law of sines. For example, if you have a triangle with side lengths $a = 3$ units, $b = 4$ units, and $c = 5$ units, you can use the law of cosines to find the length of side $d$:

$$d^2 = a^2 + b^2 - 2ab \cos(C)$$

where $C$ is the angle between sides $a$ and $b$.

Q: What is the law of cosines?

A: The law of cosines is a formula that relates the lengths of the sides of a triangle to the cosine of one of its angles. The formula is:

$$c^2 = a^2 + b^2 - 2ab \cos(C)$$

where $a$ and $b$ are the lengths of the two sides, $c$ is the length of the third side, and $C$ is the angle between sides $a$ and $b$.

Q: What is the law of sines?

A: The law of sines is a formula that relates the lengths of the sides of a triangle to the sines of its angles. The formula is:

$$\frac{a}{\sin(A)} = \frac{b}{\sin(B)} = \frac{c}{\sin(C)}$$

where $a$, $b$, and $c$ are the lengths of the sides, and $A$, $B$, and $C$ are the angles opposite those sides.

Conclusion

In this article, we provided a Q&A guide to help you understand the concepts and techniques used to solve geometric problems. We covered the Pythagorean theorem, the law of cosines, and the law of sines, and provided examples of how to apply these formulas to find the length of a side in a triangle. We hope this guide has been helpful in your studies of geometry.