Practice Problem - Sai Govindu
Proof: Given ABCD is a parallelogram; EA = DF
Prove: <E = <F
Two - Column Proof and Ending Diagram
Prove: <E = <F
Two - Column Proof and Ending Diagram
Explanation: Since we are given that ABCD is a parallelogram, we already know many things. We know that AC = BD and <C = <B because of parallelogram properties. Therefore, we can use that to prove <E = <F. We could use CPCTC to prove both of those angles congruent but if so, we need to prove the two triangle congruent first. We know AC = BD and we are given DF = EA so we already have 2 sides congruent. We either need another side congruence or angle congruence. Since we know ABCD is a parallelogram, we can use the alternate interior angles theorem to prove angles congruent. We already know <C = <D so if we use those angles and the alt. int. theorem, we can say that <C = <A and <B = <D. Then using the substitution property, we can take out or substitute <C and <B because they are equal and prove that <A = <D. Because of Side-Angle-Side congruence, we can prove triangles ACE and DBF congruent and finally prove <E = to <F using CPCTC from the 2 triangles.
Practice Problem - Sai Govindu
The diameter of this circle is 8 cm. Find the area of the shaded region.(regular hexagon) Leave answers exact.
Explanation: Since the diameter of the circle is 10cm, the radius for the regular hexagon is and of course the circle is 5cm. To solve for the shaded region, we need to first find the area of the circle and then subtract that from the area of the regular hexagon. The formula for solving the area of a circle is pi * r ^2.
Circle Area: pi * 5^2
circle area = 25pi
To find the area of the area of the regular hexagon, we can use the apothem and this formula: aP/2.
Shown below:
Circle Area: pi * 5^2
circle area = 25pi
To find the area of the area of the regular hexagon, we can use the apothem and this formula: aP/2.
Shown below:
Since we know the radius is 5 cm and it bisects angles, we can solve using this special right triangle and their properties. Half the side length of the hexagon would be 2.5 cm and the apothem would be 2.5sqrt3. Then 1 side length would be 5 and the perimeter as a result would be 30cm. Now we can just input these values into the formula. see finished picture
area = apothem*perimeter/2
area = 2.5sqrt3 * 30/2
area = 2.5sqrt3 *15
area = 37.5sqrt3 cm^2
We found that the area of the regular hexagon was 37.5sqrt3 cm^2. We also found that the area of the circumscribed circle was 25pi. To find the area of the shaded region, we subtract the area of the circle to the hexagon area. 25pi - 37.5sqrt3 cm^2
area = 2.5sqrt3 * 30/2
area = 2.5sqrt3 *15
area = 37.5sqrt3 cm^2
We found that the area of the regular hexagon was 37.5sqrt3 cm^2. We also found that the area of the circumscribed circle was 25pi. To find the area of the shaded region, we subtract the area of the circle to the hexagon area. 25pi - 37.5sqrt3 cm^2
Practice Problems- Amoagh Gopinath
![Picture](/uploads/3/8/6/0/38601155/897746.png?250)
1. There are two rectangles. The smaller rectangle's small side length is 6 inches while the larger rectangles's small side's length is 9 inches. The perimeter of the smaller rectangle is 28 inches. Find the area of the larger rectangle.
Answer:
You first have to find the ratio between the side lengths:
6:9=2:3
Then you use the perimeter and the given side length to find the missing side length:
2*6+2x=28
12+2x=28
2x=16
x=8
Next find the area of the smaller rectangle:
6*8=48
Finally use the scale factor to find the larger rectangles area:
2/3=48/x
x=72
The area of the larger rectangle is 72 in. squared
You first have to find the ratio between the side lengths:
6:9=2:3
Then you use the perimeter and the given side length to find the missing side length:
2*6+2x=28
12+2x=28
2x=16
x=8
Next find the area of the smaller rectangle:
6*8=48
Finally use the scale factor to find the larger rectangles area:
2/3=48/x
x=72
The area of the larger rectangle is 72 in. squared
![Picture](/uploads/3/8/6/0/38601155/4240429.png?250)
2. Solve the problems below using this regular hexagon with a radius of 6.
a. Find the apothem of the hexagon
b. Find the perimeter of the hexagon.
c. Find the area of the hexagon.
Answer:
a.The apothem and the radius form a 30:60:90 triangle
Which makes the apothem 3√3
b. Each side is equal to 6.
6*6=36
c.A=1/2*aP
A=1/2*3√3*36
A=3√3*18
A=53√3
a. Find the apothem of the hexagon
b. Find the perimeter of the hexagon.
c. Find the area of the hexagon.
Answer:
a.The apothem and the radius form a 30:60:90 triangle
Which makes the apothem 3√3
b. Each side is equal to 6.
6*6=36
c.A=1/2*aP
A=1/2*3√3*36
A=3√3*18
A=53√3