Out of 31 questions these are ones I struggled with. after the = sign is what I think the answer( except for #4, I don't know what to do at all), is so if I'm wrong can someone correct it with an explanation/formula with it please?
1.) Find the area of the parallelogram. ( base(s)=10cm, side length(s)=6cm, height=? and the angle of elevation=60'
a.)15√3 cm^2
b.)30√3 cm^2
c.)15√2 cm^2
=30√3 cm^2
2.) What is the length of the radius of a circle with the center at O(-3,4) and passes through the point P(2,-5)?
a.)9
b.)12.5
c.)10.3
=9
3.) What is the equation for a circle with a diameter having the following endpoints: N(0,1) and Q(18,1)?
a.)(x - 9)^2 + (y - 1)^2 = 81
b.)(x)^2 + (y)^2 = 72
c.)(x + 2)^2 + (y - 7)^2 = 56
=(x)^2 + (y)^2 = 72
4.) Using coordinate geometry to prove that the diagonals of a square are perpendicular to each other.
Given: Vertices are at A(0,0), B(a,0), C(a,a) and D(0,a) Slope of AC=1; Slope of BD=-1
Type your proof into the text box below.
=
1.) To find the area of a parallelogram, you can use the formula:
Area = base * height
In this case, the base of the parallelogram is 10cm. The height is the distance from the base to the opposite side, which can be found using the side length and the angle of elevation.
Since the angle of elevation is given as 60 degrees, we can use trigonometry to find the height. The height is the side length multiplied by the sine of the angle of elevation.
Therefore, the height = 6cm * sin(60 degrees) = 6cm * √3 / 2 = 3√3 cm
Substituting the values into the area formula:
Area = 10cm * 3√3 cm
Area = 30√3 cm^2
So the correct answer is b) 30√3 cm^2.
2.) The distance between the center of a circle and a point on its circumference is equal to the radius of the circle.
To find the distance between points O(-3,4) and P(2,-5), you can use the distance formula:
Distance = √((x2 - x1)^2 + (y2 - y1)^2)
Substituting the coordinates of O and P into the formula:
Distance = √((2 - (-3))^2 + (-5 - 4)^2)
Distance = √((2 + 3)^2 + (-5 - 4)^2)
Distance = √(5^2 + (-9)^2)
Distance = √(25 + 81)
Distance = √106
So, the length of the radius is √106. The closest whole number to √106 is 10, so the correct answer is c) 10.3.
3.) The equation of a circle with the center at (h, k) and radius r is given by:
(x - h)^2 + (y - k)^2 = r^2
In this case, the center of the circle is the midpoint of the diameter, which can be calculated as the average of the x-coordinates and average of the y-coordinates of the endpoints.
The x-coordinate of the center = (0 + 18) / 2 = 9
The y-coordinate of the center = (1 + 1) / 2 = 1
The radius of the circle is half the length of the diameter, which is the distance between the endpoints N(0,1) and Q(18,1).
Using the distance formula:
Radius = √((18 - 0)^2 + (1 - 1)^2)
Radius = √(324)
Radius = 18
Substituting the values into the circle equation:
(x - 9)^2 + (y - 1)^2 = 18^2
(x - 9)^2 + (y - 1)^2 = 324
So the correct answer is a) (x - 9)^2 + (y - 1)^2 = 81.
4.) To prove that the diagonals of a square are perpendicular to each other using coordinate geometry, you need to show that the product of their slopes is -1.
Given the vertices A(0,0), B(a,0), C(a,a), and D(0,a), we can calculate the slopes of the diagonals AC and BD.
The slope of AC = (a - 0) / (a - 0) = 1
The slope of BD = (0 - a) / (0 - a) = -1
Since the product of the slopes is 1 * -1 = -1, the diagonals are perpendicular to each other.
Therefore, the proof is complete.