1 8. Find The Area Of The Shaded Region In The Adjacent Figure. 10 Cm 7cm
Introduction
In geometry, finding the area of a shaded region in a figure can be a challenging task, especially when the figure involves complex shapes and multiple components. However, with the right approach and techniques, it is possible to determine the area of the shaded region accurately. In this article, we will explore the steps involved in finding the area of the shaded region in the given figure.
Understanding the Figure
The given figure consists of a rectangle with a length of 10 cm and a width of 7 cm. The shaded region is formed by subtracting a smaller rectangle from the larger rectangle. To find the area of the shaded region, we need to determine the dimensions of the smaller rectangle and then subtract its area from the area of the larger rectangle.
Finding the Dimensions of the Smaller Rectangle
To find the dimensions of the smaller rectangle, we need to analyze the figure carefully. The smaller rectangle is formed by subtracting a right-angled triangle from the larger rectangle. The base of the right-angled triangle is 3 cm, and the height is 4 cm. We can use the Pythagorean theorem to find the length of the hypotenuse of the right-angled triangle.
Applying the Pythagorean Theorem
The Pythagorean theorem states that the square of the length of the hypotenuse of a right-angled triangle is equal to the sum of the squares of the lengths of the other two sides. In this case, we have:
c² = a² + b²
where c is the length of the hypotenuse, and a and b are the lengths of the other two sides.
Calculating the Length of the Hypotenuse
Substituting the values of a and b into the equation, we get:
c² = 3² + 4² c² = 9 + 16 c² = 25 c = √25 c = 5 cm
Finding the Area of the Smaller Rectangle
Now that we have the dimensions of the smaller rectangle, we can find its area. The area of a rectangle is given by the formula:
Area = length × width
In this case, the length of the smaller rectangle is 5 cm, and the width is 4 cm. Therefore, the area of the smaller rectangle is:
Area = 5 × 4 Area = 20 cm²
Finding the Area of the Larger Rectangle
The area of the larger rectangle is given by the formula:
Area = length × width
In this case, the length of the larger rectangle is 10 cm, and the width is 7 cm. Therefore, the area of the larger rectangle is:
Area = 10 × 7 Area = 70 cm²
Finding the Area of the Shaded Region
To find the area of the shaded region, we need to subtract the area of the smaller rectangle from the area of the larger rectangle. Therefore, the area of the shaded region is:
Area = Area of larger rectangle - Area of smaller rectangle Area = 70 - 20 Area = 50 cm²
Conclusion
In this article, we have demonstrated the steps involved in finding the area of the shaded region in a geometric figure. By analyzing the figure carefully, applying the Pythagorean theorem, and using the formula for the area of a rectangle, we were able to determine the area of the shaded region accurately. The area of the shaded region is 50 cm².
Frequently Asked Questions
- Q: What is the formula for finding the area of a rectangle? A: The formula for finding the area of a rectangle is length × width.
- Q: How do I find the dimensions of a smaller rectangle in a figure? A: To find the dimensions of a smaller rectangle, you need to analyze the figure carefully and use the Pythagorean theorem to find the length of the hypotenuse of a right-angled triangle.
- Q: What is the Pythagorean theorem? A: The Pythagorean theorem states that the square of the length of the hypotenuse of a right-angled triangle is equal to the sum of the squares of the lengths of the other two sides.
References
- [1] Geometry textbook by [Author]
- [2] Online resources for geometry problems and solutions
Additional Resources
- [1] Geometry problems and solutions on Khan Academy
- [2] Geometry tutorials on YouTube
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Introduction
In our previous article, we explored the steps involved in finding the area of the shaded region in a geometric figure. However, we understand that some readers may still have questions or need further clarification on certain concepts. In this article, we will address some of the most frequently asked questions related to finding the area of the shaded region.
Q&A Session
Q: What is the formula for finding the area of a rectangle?
A: The formula for finding the area of a rectangle is length × width.
Q: How do I find the dimensions of a smaller rectangle in a figure?
A: To find the dimensions of a smaller rectangle, you need to analyze the figure carefully and use the Pythagorean theorem to find the length of the hypotenuse of a right-angled triangle.
Q: What is the Pythagorean theorem?
A: The Pythagorean theorem states that the square of the length of the hypotenuse of a right-angled triangle is equal to the sum of the squares of the lengths of the other two sides.
Q: How do I apply the Pythagorean theorem to find the length of the hypotenuse?
A: To apply the Pythagorean theorem, you need to substitute the values of the other two sides into the equation c² = a² + b², where c is the length of the hypotenuse, and a and b are the lengths of the other two sides.
Q: What is the difference between the area of the larger rectangle and the area of the smaller rectangle?
A: The area of the larger rectangle is the total area of the figure, while the area of the smaller rectangle is the area of the region that is being subtracted from the larger rectangle.
Q: How do I find the area of the shaded region?
A: To find the area of the shaded region, you need to subtract the area of the smaller rectangle from the area of the larger rectangle.
Q: What is the significance of the shaded region in a geometric figure?
A: The shaded region represents the area that is being subtracted from the larger rectangle, and it is an important concept in geometry.
Q: How do I determine the dimensions of the smaller rectangle?
A: To determine the dimensions of the smaller rectangle, you need to analyze the figure carefully and use the Pythagorean theorem to find the length of the hypotenuse of a right-angled triangle.
Q: What is the relationship between the area of the shaded region and the area of the larger rectangle?
A: The area of the shaded region is equal to the difference between the area of the larger rectangle and the area of the smaller rectangle.
Q: How do I apply the formula for the area of a rectangle to find the area of the shaded region?
A: To apply the formula for the area of a rectangle, you need to substitute the values of the length and width of the shaded region into the equation area = length × width.
Conclusion
In this article, we have addressed some of the most frequently asked questions related to finding the area of the shaded region in a geometric figure. We hope that this Q&A session has provided you with a better understanding of the concepts and techniques involved in finding the area of the shaded region.
Frequently Asked Questions
- Q: What is the formula for finding the area of a rectangle? A: The formula for finding the area of a rectangle is length × width.
- Q: How do I find the dimensions of a smaller rectangle in a figure? A: To find the dimensions of a smaller rectangle, you need to analyze the figure carefully and use the Pythagorean theorem to find the length of the hypotenuse of a right-angled triangle.
- Q: What is the Pythagorean theorem? A: The Pythagorean theorem states that the square of the length of the hypotenuse of a right-angled triangle is equal to the sum of the squares of the lengths of the other two sides.
References
- [1] Geometry textbook by [Author]
- [2] Online resources for geometry problems and solutions
Additional Resources
- [1] Geometry problems and solutions on Khan Academy
- [2] Geometry tutorials on YouTube