Introduction
Welcome to our comprehensive guide on 11.2 Practice B Geometry Answers. In this article, we will go through the various questions and solutions provided in Practice B of Chapter 11.2 in Geometry. Whether you are a student looking for help or a teacher searching for additional resources, this article is designed to provide you with a clear understanding of the concepts covered in these practice exercises. Let's dive in!
Question 1: Determining the Length of a Chord
In this question, you are given a circle with a radius of 8 units and a chord that is 10 units long. Your task is to find the distance between the chord and the center of the circle. To solve this problem, we can use the Pythagorean theorem and the properties of right triangles. Let's break it down step by step:
Step 1: Draw the Diagram
Start by drawing a circle with a radius of 8 units. Then, draw a chord that is 10 units long. Label the center of the circle as point O and the midpoint of the chord as point M.
Step 2: Identify the Right Triangle
Since the chord is perpendicular to the radius at the point of intersection, we can identify a right triangle formed by the radius, the distance between the chord and the center, and half the length of the chord. Let's call the distance between the chord and the center as x.
Step 3: Apply the Pythagorean Theorem
Using the Pythagorean theorem, we can write the equation:
x^2 + (10/2)^2 = 8^2
Simplifying this equation, we get:
x^2 + 25 = 64
x^2 = 39
Therefore, x = √39 units.
Question 2: Finding the Area of a Sector
In this question, you are given a sector of a circle with a central angle of 60 degrees and a radius of 12 units. Your task is to find the area of the sector. To solve this problem, we can use the formula for the area of a sector. Let's break it down step by step:
Step 1: Identify the Given Values
From the question, we know that the central angle is 60 degrees and the radius is 12 units. We need to use these values to find the area of the sector.
Step 2: Convert the Central Angle to Radians
In order to use the formula for the area of a sector, we need to convert the central angle from degrees to radians. Since there are π radians in 180 degrees, we can write:
60 degrees * (π/180) = (π/3) radians
Step 3: Apply the Formula
The formula for the area of a sector is:
Area = (θ/360) * π * r^2
Plugging in the values we have:
Area = ((π/3)/360) * π * (12)^2
Area = (π/180) * π * 144
Simplifying this expression, we get:
Area = (π^2 * 144)/180
Therefore, the area of the sector is (π^2 * 144)/180 square units.
Question 3: Finding the Equation of a Circle
In this question, you are given a circle with its center at (-2, 3) and a point on the circle at (4, -1). Your task is to find the equation of the circle. To solve this problem, we can use the equation of a circle. Let's break it down step by step:
Step 1: Identify the Given Values
From the question, we know that the center of the circle is (-2, 3) and a point on the circle is (4, -1). We need to use these values to find the equation of the circle.
Step 2: Apply the Formula
The equation of a circle with center (h, k) and radius r is:
(x - h)^2 + (y - k)^2 = r^2
Plugging in the values we have:
(x - (-2))^2 + (y - 3)^2 = r^2
(x + 2)^2 + (y - 3)^2 = r^2
Step 3: Calculate the Radius
To find the radius, we can use the distance formula between the center of the circle and a point on the circle. Plugging in the values, we get:
r = √[(4 - (-2))^2 + (-1 - 3)^2]
r = √[6^2 + (-4)^2]
r = √[36 + 16]
r = √52
Step 4: Final Equation of the Circle
Substituting the value of r into the equation, we get:
(x + 2)^2 + (y - 3)^2 = (√52)^2
(x + 2)^2 + (y - 3)^2 = 52
Therefore, the equation of the circle is (x + 2)^2 + (y - 3)^2 = 52.
Question 4: Finding the Length of an Arc
In this question, you are given a circle with a radius of 10 units and a central angle of 45 degrees. Your task is to find the length of the arc intercepted by this central angle. To solve this problem, we can use the formula for the length of an arc. Let's break it down step by step:
Step 1: Identify the Given Values
From the question, we know that the radius of the circle is 10 units and the central angle is 45 degrees. We need to use these values to find the length of the arc.
Step 2: Convert the Central Angle to Radians
To use the formula for the length of an arc, we need to convert the central angle from degrees to radians. Since there are π radians in 180 degrees, we can write:
45 degrees * (π/180) = (π/4) radians
Step 3: Apply the Formula
The formula for the length of an arc is:
Length = (θ/360) * 2 * π * r
Plugging in the values we have:
Length = ((π/4)/360) * 2 * π * 10
Length = (π/1440) * 2 * π * 10
Simplifying this expression, we get:
Length = (2π^2 * 10)/1440
Therefore, the length of the arc intercepted by the central angle of 45 degrees is (2π^2 * 10)/1440 units.
Question 5: Determining the Perpendicular Bisector of a Line Segment
In this question, you are given a line segment AB with endpoints A(-3, 2) and B(3, 6). Your task is to find the equation of the perpendicular bisector of this line segment. To solve this problem, we can use the midpoint formula and the negative reciprocal of the slope. Let's break it down step by step:
Step 1: Identify the Given Values
From the question, we know the coordinates of the endpoints A(-3, 2) and B(3, 6). We need to use these values to find the equation of the perpendicular bisector.
Step 2: Find the Midpoint
The midpoint of a line segment with endpoints (x1, y1) and (x2, y2) is given by the formula:
(x, y) = ((x1 + x2)/2, (y1 + y2)/2)
Plugging in the values we have:
(x, y) = ((-3 + 3)/2, (2 + 6)/2)
(x, y) = (0, 4)
Step 3: Find the Slope of the Line Segment
The slope of a line segment with endpoints (x1, y1) and (x2, y2) is given by the formula:
m = (y2 - y1)/(x2 - x1)
Plugging in the values we have:
m = (6 - 2)/(3 - (-