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.
=

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To find the area of a parallelogram, you can use the formula A = base x height, where the base is one of the side lengths and the height is the perpendicular distance between the base and the opposite side. In this case, the base length is given as 10 cm.

To find the height, you can use the angle of elevation and the side length. The height is the length of the side multiplied by the sine of the angle. In this case, the side length is given as 6 cm and the angle of elevation is 60 degrees (converted to radians).

So, height = 6 cm x sin(60 degrees) = 6 cm x √3/2 = 3√3 cm.

Now, you can substitute the base length and height into the formula to find the area:
A = 10 cm x 3√3 cm = 30√3 cm^2.

Therefore, the correct answer is option b) 30√3 cm^2.

For the second question:
To find the length of the radius of a circle with the center at O(-3,4) and passes through the point P(2,-5), you can use the distance formula.

The distance formula is given by:
d = √((x2 - x1)^2 + (y2 - y1)^2), where (x1, y1) and (x2, y2) are the coordinates of the two points.

In this case, the coordinates of O are (-3,4) and the coordinates of P are (2,-5).

Using the distance formula, the distance between O and P is:
d = √((2 - (-3))^2 + (-5 - 4)^2)
= √((2 + 3)^2 + (-5 - 4)^2)
= √(5^2 + (-9)^2)
= √(25 + 81)
= √106

Therefore, the length of the radius is √106, which is approximately 10.3.

So, the correct answer is option c) 10.3.

For the third question:
To find the equation of a circle given the diameter with endpoints N(0,1) and Q(18,1), you can use the formula (x - h)^2 + (y - k)^2 = r^2, where (h, k) is the center of the circle and r is the radius.

In this case, the diameter is the distance between N and Q, which is 18 - 0 = 18.

The center of the circle is the midpoint of the diameter, which is (0 + 18)/2 = 9.

The radius is half of the diameter, which is 18/2 = 9.

Now, you can substitute the values into the formula to get the equation of the circle:
(x - 9)^2 + (y - 1)^2 = 9^2
(x - 9)^2 + (y - 1)^2 = 81

Therefore, the correct answer is option a) (x - 9)^2 + (y - 1)^2 = 81.

For the fourth question:
To prove that the diagonals of a square are perpendicular to each other using coordinate geometry, you need to show that the slopes of the diagonals are negative reciprocals of each other.

Given square ABCD with vertices A(0,0), B(a,0), C(a,a), and D(0,a), the slope of diagonal AC can be found using the formula for slope: m = (y2 - y1)/(x2 - x1).

The coordinates of points A and C are (0,0) and (a,a) respectively, so the slope of AC can be calculated as (a - 0)/(a - 0) = 1.

Similarly, the slope of diagonal BD can be found using the coordinates of points B and D, which are (a,0) and (0,a) respectively. The slope of BD can be calculated as (0 - a)/(a - 0) = -1.

Since the slope of diagonal AC is 1 and the slope of diagonal BD is -1, we can see that they are indeed negative reciprocals of each other.

Therefore, the diagonals of a square are perpendicular to each other.

Explanation: To find the slope of a line, you can use the formula m = (y2 - y1)/(x2 - x1), where (x1, y1) and (x2, y2) are coordinates of two points on the line. By comparing the slopes of the two diagonals of the square, we can determine if they are perpendicular to each other.