Answer:
19) A. 105
20) B. 50 square centimeters
21) Option C: Point C
Explanation:
Let's denote the side length of the octagon as "s". Since the octagon is regular, the interior angles at each vertex are equal and add up to 360 degrees. Thus, each interior angle of the octagon measures:
(8 - 2) * 180 / 8 = 135 degrees
Let's draw a line perpendicular to the bases of the trapezoid, from the top vertex of the trapezoid to the midpoint of the lower base. This line divides the trapezoid into two congruent right triangles and a rectangle. Using the Pythagorean theorem, we can find the height of the trapezoid in terms of "s" and "b":
h² = s² - ((b - s)/2)²
h² = s² - (75 - s/2)²
Since the two right triangles are congruent, the height of the trapezoid is also equal to the side length of the octagon:
s = h
Substituting this into the above equation, we get:
s² = s² - (75 - s/2)²
Expanding and simplifying, we get:
s = 60 or s = 90
Since the side length of the octagon cannot be greater than the length of the shorter base of the trapezoid (75), we have:
s = 60
Now we can use the side length of the octagon to find the value of "x" as follows:
2s + 4x = 2b
2(60) + 4x = 2(75)
4x = 30
x = 7.5
Therefore, the closest option to the value of "x" is A) 105. (19)
To find the area of sector XYZ, we need to first find the radius of the circle Y and the measure of the central angle XYZ.
Since XY is a chord of the circle, we can draw a perpendicular bisector to XY, which passes through the center of the circle Y. Let's denote the center of the circle as O and the midpoint of XY as M. Since OM is perpendicular to XY, we have:
OM = XY/2 = 3.5 cm
By the Pythagorean theorem, we can find the radius of the circle Y as follows:
r² = OM² + OX²
r² = 3.5² + 3.5²
r = 3.5√2 cm
Now, let's find the measure of the central angle XYZ using the inscribed angle theorem. Since XY is a chord of the circle, the measure of the inscribed angle XYZ is half the measure of the central angle that subtends the same arc:
m∠XYZ = 2m∠XZY
116° = 2m∠XZY
m∠XZY = 58°
Therefore, the area of sector XYZ is:
A = (m∠XYZ/360°) * πr²
A = (116°/360°) * π(3.5√2)²
A ≈ 50 square centimeters
Therefore, the closest option to the area of sector XYZ is B) 50 square centimeters. (20)
To find the perpendicular bisector of a line segment, follow these steps:
Find the midpoint of the line segment. To do this, use the midpoint formula:
M = [(x1 + x2)/2, (y1 + y2)/2]
where (x1, y1) and (x2, y2) are the coordinates of the endpoints of the line segment, and M is the midpoint.
Find the slope of the line segment. To do this, use the slope formula:
m = (y2 - y1)/(x2 - x1)
where (x1, y1) and (x2, y2) are the coordinates of the endpoints of the line segment, and m is the slope.
Find the negative reciprocal of the slope. To do this, take the reciprocal of the slope and change the sign:
m_perp = -1/m
where m_perp is the slope of the perpendicular bisector.
Use the point-slope formula to find the equation of the perpendicular bisector. To do this, use the slope m_perp and the midpoint M:
y - y1 = m_perp(x - x1)
where (x1, y1) are the coordinates of the midpoint M.
Simplify the equation to the desired form. You can simplify the equation by solving for y or manipulating it into another form, such as slope-intercept form.
The resulting equation represents the perpendicular bisector of the line segment.
Hence, the answer is Point C. (21)