## In the Figure, What is the Value of x?

In geometry, we are often tasked with finding the values of unknown variables, such as angles or side lengths, based on given information. One common problem involves determining the value of x in various figures, such as triangles or circles. Understanding the relationships between different parts of a figure is crucial for solving these types of problems.

### Pain Points

When trying to find the value of x in a figure, it can be frustrating to:

- Lack a clear understanding of the relationships between the known and unknown values.
- Make careless mistakes in calculations or interpretations.
- Misinterpret the given information, leading to incorrect conclusions.

### Finding the Value of x

In the figure shown, we are given a triangle with sides of length 5, 12, and x. We need to find the value of x. Since the sum of the angles in a triangle is always 180 degrees, we can use angle relationships to solve for x.

**Step 1:** Label the angles opposite each side, using the Law of Sines.

```
∠A / 5 = ∠B / 12 = ∠C / x
```

**Step 2:** Use the given angle measures to simplify the equation.

```
∠A = 30°, ∠B = 90°, ∠C = 60°
30° / 5 = 90° / 12 = 60° / x
```

**Step 3:** Solve for x by cross-multiplying and dividing.

```
30° * 12 = 90° * 5
360° = 450°
x = 450° / 60°
x = 7.5
```

Therefore, the value of x in the given figure is 7.5.

### Summary

Understanding the relationships between angles and sides in a figure is essential for solving problems involving unknown values like x. By using the Law of Sines and carefully applying the given information, we can find the value of x accurately, avoiding common pain points like incorrect assumptions or calculation errors.

**Understanding the Value of X in the Given Figure**

**Introduction**

The given figure presents an intriguing mathematical puzzle that involves determining the value of the variable x. This article delves into the intricate intricacies of this puzzle, exploring the steps and concepts necessary for unraveling its solution.

**Exploring the Puzzle**

**Given Information**

The provided figure showcases three circles, each with a different diameter. The diameter of the largest circle is given as 12 units, while the diameters of the two smaller circles are labeled as x and y units, respectively. Additionally, the figure indicates that the sum of the areas of the two smaller circles is equal to the area of the largest circle.

**Relationship Between Circles**

The area of a circle is directly proportional to the square of its radius. Therefore, the following equation can be established:

```
πr² = Area
```

where:

- π ≈ 3.14
- r = Radius of the circle

**Expressing Radii in Terms of Diameters**

The radius of a circle is half of its diameter. Thus, we can express the radii of the circles in terms of their diameters:

- Radius of the largest circle = 12/2 = 6 units
- Radius of the smaller circles = x/2 and y/2 units

**Equating Areas**

The sum of the areas of the smaller circles is given as the area of the largest circle. Substituting the area formula and expressing radii in terms of diameters, we get:

```
π(x/2)² + π(y/2)² = π(6)²
```

Simplifying the equation:

```
(x²/4 + y²/4) = 36
```

```
x² + y² = 144
```

**Solving for x**

To isolate x, we need to subtract y² from both sides of the equation:

```
x² = 144 - y²
```

Since x is positive, taking the square root yields:

```
x = √(144 - y²)
```

**Conclusion**

The value of x in the given figure depends on the value of y. However, without further information about y, we cannot determine the exact value of x.

**Frequently Asked Questions (FAQs)**

**1. What is the relationship between the areas of the circles?**

The sum of the areas of the two smaller circles is equal to the area of the largest circle.

**2. How is the value of x determined?**

The value of x is derived from the equation x² + y² = 144, where y represents the diameter of the other smaller circle.

**3. Can we determine the exact value of x without knowing y?**

No, the exact value of x cannot be determined unless the value of y is known.

**4. What is the significance of the radii in solving the puzzle?**

The radii of the circles are used to express their areas in terms of the diameters given in the figure.

**5. Is there any other information that would help solve the puzzle?**

Additional information, such as the value of y or the ratio between the diameters of the circles, would assist in finding the exact value of x.

.

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