Have you ever struggled with solving math problems related to the relative positions of lines and circles? Well, worry no more! In this article, we will delve into the theory behind this topic and provide you with step-by-step solutions. Our goal is to help you understand the concepts better and equip teachers with valuable resources for instructing students effectively.
I. Reviewing the Fundamentals of Math Lesson 20 Page 110 Workbook 9
Before tackling any exercise, it is crucial to have a solid grasp of the general knowledge pertaining to the specific problem type and the overall lesson. This lesson specifically focuses on the relative positions of lines and circles, as well as the application of this knowledge to various problem scenarios. So, let’s take a moment to revisit the theory before diving into the detailed solutions for Exercise 20 on page 110 of Workbook 9.
1. Key Concepts to Remember
- The relative positions of lines and circles
- Given a circle (O;R) and an arbitrary line Δ, the distance from the center O of the circle to the line is represented as d.
Case 1: Intersection of Line Δ and Circle (O;R)
In this case, the line and the circle have two points of intersection, and the distance d is less than the radius R.
Case 2: Tangent Line Δ and Circle (O;R)
Here, the line and the circle share one point of contact, and the distance d is equal to the radius R. The line Δ is referred to as a tangent line, while point B denotes the point of tangency.
Case 3: Non-intersecting Line Δ and Circle (O;R)
In this scenario, the line and the circle do not intersect, and the distance d is greater than the radius R. This gives us a clear understanding of the relative positions of lines and circles.
2. Common Problem Types
Let’s explore some typical problem types associated with this lesson.
Problem Type 1: Determining the Relative Positions of Lines and Circles
Method:
- Refer to the relative positions table
Problem Type 2: Calculating Lengths Based on Tangent Properties
Method:
- Utilize the tangent property and the Pythagorean theorem.
Problem Type 3: Finding Sets of Points Meeting Given Conditions
Method:
- Utilize the angle bisector property and equally spaced parallel lines to identify the desired set of points.
Now that we have reviewed the necessary theory, let’s proceed to the in-depth solutions for Exercise 20 on page 110 of Workbook 9.
II. Step-by-Step Solutions for Exercise 20 on Page 110 of Workbook 9
After understanding the theoretical aspects covered above, you should now have a clearer picture of the relative positions of lines and circles. So, let’s dive into the specific solution for Exercise 20 on page 110 of Workbook 9!
Problem Statement:
Given a circle with center O and a radius of 6 cm, and a point A located 10 cm away from O. We draw tangent line AB with point B touching the circle. Find the length of AB.
Solution:
Consider the circle (O).
We have:
- Since B is the point of tangency, OB = R = 6 cm.
- AB is the tangent line at B, so AB ⊥ OB at B.
- In right triangle ABO (as AB ⊥ OB), we can apply the Pythagorean theorem.
- OA^2 = OB^2 + AB^2
- Therefore, AB^2 = OA^2 – OB^2 = 10^2 – 6^2 = 64.
III. Additional Exercise Hints – Workbook 9 Page 110
With the specific solution provided for Exercise 20 on page 110 of Workbook 9, you should now have a better grasp of the method and how to approach similar problems. To reinforce your understanding, let’s explore a few more exercises from this workbook.
Exercise 17 (Workbook 9 Page 109)
Fill in the blanks (…) in the following table (R represents the radius of the circle, and d represents the distance from the center to the line):
Solution:
Referring to the relation between d and R, we can complete the table as follows:
Exercise 18 (Workbook 9 Page 110)
On the coordinate plane Oxy, given the point A(3; 4). Determine the relative positions of the circle (A; 3) and the coordinate axes.
Solution:
We draw AH ⊥ Ox and AK ⊥ Oy.
Since AH = 4 > R = 3, the circle center (A) and the horizontal axis do not intersect.
Since AK = 3 = R, the circle center (A) and the vertical axis are tangent to each other.
Exercise 19 (Workbook 9 Page 110)
Given the line xy, on the coordinate plane. On which line are the centers of the circles with a radius of 1cm, tangent to xy?
Solution:
Let O be the center of an arbitrary circle with a radius of 1cm, which is tangent to the line xy.
We have: R = 1, and the circle is tangent to the line xy, so d = R. Therefore, d = 1.
As a result, the centers of the circles are located on lines (a) and (b), which are parallel to xy and 1cm away from it.
IV. Conclusion
By now, you should have acquired a solid understanding of the relative positions of lines and circles, thanks to the information we have shared throughout this engaging lesson.
The exercises on page 110 of Workbook 9, especially Exercise 20, have been thoroughly explained and accompanied by meticulous step-by-step solutions and applicable formulas.
We sincerely hope that this knowledge will serve as a valuable asset in mastering your math skills, specifically in the realm of Grade 9. Make sure to stay tuned to Kiến Guru for more enlightening articles and comprehensive problem-solving guides across various subjects.
Goodbye for now, and see you in our next exciting lessons!
