Act Prep Geometry Quiz #1

Explanation
We can see that each time, the term is being divided by - 4. Therefore, to find the next term, we need to divide the last term (-1/4) by 4. This should give us 1/16.

Explanation
The area of a circle is A=πr2.  Since the diameter is 24, the radius of the circle is 12.  Plugging 12 in for r in our equation for area, we get 144π.  However, we do not need to know the area of the whole pizza, we only want the area for one-eighth of the pizza, so we need to divide the total area by 8.  This gives us 18π.

Explanation
The area of a rectangle is length times width. This means the area of the rectangular garden is 144 square feet. The new square garden need to have the same area. Remember that all of the sides are equal in a square, so what perfect square gives us an area of 144 square feet? We can see that this number is 12, so each side of the square garden is 12 feet long.

Explanation
Remember that Pythagorean Theorem (a2+b2=c2 where c is the hypotenuse) tells us about the side lengths of a right triangle.  Therefore, a2+32=62.  Solving for a, one should get √27.  However, we want this answer in simplest radical form, so the final answer is 3√3.

Explanation
Remember that the area of the triangle is base times height divided by 2. So the area of triangle DGF minus the area of triangle EGF should give us the area of triangle DEG. The area of the triangle DGF is (19*10)/2=95 and the area of triangle EGF is (7*10)/2=35. We know that 95-35=60, so the area of triangle DEG is 60 square units.

Explanation
We need to find the area of the larger circle and subtract out the area of the smaller circle. Remember that the area of a circle is A=πr2.  We know the radius of the smaller circle is 5 and the area of the bigger circle is the diameter of the smaller circle, so the radius of the bigger circle is 10.  This means the area of the bigger circle is 100π and the area of the smaller circle is 25π.  So 100π - 25π = 75π, which is the area left over in square units.

Explanation
If we count the pieces of fencing that are completely horizontal and vertical, we see that there are 20 posts. Since each post is 10 feet, we know that the perimeter should be more than 200 feet, since we have not counted the length of that diagonal post. Since the post we haven't considered yet is a diagonal piece, we know the length of this piece should be more than 10 feet. Therefore, the perimeter should be greater than 210 feet.

Explanation
In order to find the perimeter of WXZY, we need to know each side length. Note that each side length is equal to the diameter of two circles and each diameter is double the length of the radius. Therefore, each side is as long as four radii. We know that the area of one circle is 4π and the equation for the area of a circle is A=πr2. Therefore, 4π=πr2.  Solving for r gives us r=2.  Therefore, the length of one side is 8.  Since there are four equal sides, the perimeter is 32.

Explanation
By sketching a plot of the points, we can see that each side has a length of 7 units. Therefore, we need to place another point 7 units below the point (3, 4). We can see that this point is (4, -4) and this point completes the square.

Explanation
Since M is the midpoint, we know that RM and MS have to be equal and RM+MS=RS. Therefore, plugging in what we know, 3x+1+3x+1=38. Simplifying gives us, 6x+2=38. Solving for x gives us x=6.