MCHS ACT Review

1
MCHS ACT Review

• Plane Geometry

2
Created by Pam Callahan

• Spring 2013 Edition

3
Angle Measures
4
Solution

• There are two facts to know here all triangles
  add to 180 degrees and all lines add to 180
  degrees. You are finding Angle ABC, which is
  outside the two triangles. Angle ABD is 60
  degrees. Angle CBE is 50 degrees. These two
  angles sum to 110 degrees. This makes Angle ABC
  70 degrees because the three angles form line DE,
  which is 180 degrees. The correct answer is K.

5
Area of Rectangles Triangles
6
Solution

• Square feet means that we are looking for area.
  The front and rear are rectangles, so area
  length times width. The two sides are triangles,
  so area of those is ½ base height.
• Front 7 5 35
• Read 7 5 35
• Left side ½ 4 6 12
• Right side ½ 4 6 12
• Sum of all areas 35 35 12 12 94, so the
  correct answer is J.

7
Area and diameter
8
Solution

• Area of a circle is Apiradius squared. Here,
  we are looking for diameter, so I am using the
  second formula. Since the area is 16pi, I know
  from the formula that the radius is 4 and the
  diameter is 8, so the correct answer is C.

9
Arcs and Circles
10
Solution

• The distance from M to N represents half the
  circle, or 180 degrees. Remember that all
  circles are 360 degrees around. Angle MOP
  represents a central angle, so arc MP is the same
  measure of 60 degrees. Arc MP is what percent of
  arc MN? 60/180 1/3 or 33 1/3 , answer D.

11
Geometric Probability
12
Solution

• If he needs at least 30 points, the dart must hit
  either the 30, 40, or 50. This section has a
  radius of 6 inches (2 for each point value). We
  dont need the area with 10 or 20. We start by
  finding the area of the entire circle, which has
  a radius of 10 inches. Area piradius2
  100pi. The area of the circle which contains
  30, 40, and 50 has a radius of 6 and an area of
  36pi. Divide 36pi/100pi, the pis cancel out,
  and the area is .36 or 36. The correct answer
  is A.

13
45-45-90 Special Right Triangle
14
Solution

• All equilateral triangles have all sides equal,
  so sides AB and AC are 7 units. Triangle ACD is
  a 45-45-90, triangle, so it has a ratio of
  1-1-v2. Since the sides are 7, the hypotenuse
  must be 7v2, which is answer choice G.

15
Volume of Cubes
16
Solution

• Volumes of cubes are sides3. The original cube
  has sides of x, so the volume is xxx or x3.
  The edges of the large cube is 2x, so its volume
  is 2x2x2x or 8x3. Divide the two volumes
  (8x3/x3) is 8 times. The correct answer is D.

17
Isosceles Triangles
18
Solution

• All triangles add to 180 degrees, and there are
  two angles not known. 180 50 130. The two
  remaining angles sum to 130, and since the
  triangle is isosceles, the angles have the same
  measure of 65 because the angles opposite the
  sides are equal. The correct answer is J.

19
Area of Circles
20
Solution

• The area of a circle is Apiradius2. The
  smaller circle has a radius of 4.5, the larger
  circle a radius of 9. To find the shaded region,
  you have to find the subtract the areas of the
  two circles.
• Large pi92 254.47 Small pi4.52 63.62
• Difference of 190.85, which is answer choice D.

21
Complementary Angles
22
Solution

• The two angles are complementary, we know they
  total 90 degrees.
• 90 (x 15) 90 x -15 when we distribute the
  negative sign through the parenthesis.
• Combine like terms, 90-15 75 x
• Correct answer is D.

23
Circles
24
Solution

• All circles have a degree measure of 360.
• All clocks have 12 numbers.
• 360/12 30 degrees, the measure between each
  number.
• If the hand moves five places, we have 5 30
  150.
• Correct answer is B.

25
Volume of Pyramid
26
Solution

• Length of base width of base is the area of base
  which is 49.
• (1/3) 49 63 (height) 1029
• Correct answer is J.

27
Quadrilaterals
28
Solution

• All quadrilaterals have angles that sum to 360
  degrees. This is found by (number of sides 2)
  180.
• Add the three angles you have given 100 75
  65 240
• 360-240 120
• The correct answer is F.

29
Area of Rectangles
30
Solution

• For the side that is 24 inches, if each tile
  covers 4 inches square, you need 6 tiles for this
  side 24/4 6
• For the other side, 64/4 is 16.
• Multiply the two 16 6 to get 96 tiles to cover
  the area. The correct answer is C.

31
Volume of Rectangular Prism
32
Solution

• The volume of any rectangular prism is length
  width height.
• Multiply 3 2 2 12
• Correct answer is G.

33
Area of Trapezoid
34
Solution

• The formula for the area of a trapezoid isA
  .5 height (b1 b2)
• A .5 (4)(7 5) 24
• Correct answer is G.

35
Pythagorean Theorem
36
Solution

• This is using the Pythagorean Theorem, A2 B2
  C2 OR
• The ACT program, option 1, option 2, will compute
  this for you.
• Since you are looking for the hypotenuse, you
  know 2 legs, A and B.
• The correct answer is G.

37
Triangle Inequality
38
Solution

• The rule is that the sum of any two sides must be
  greater than the third side, or you have a gap
  and cant construct the triangle.
• Let the third side be x and look at three
  inequalities
• X 4.5 gt 7.5 so x gt 3.5
• X 7.5 gt 4.5 so x gt -3.5 (cant use this one)
• 7.5 4.5 gt x so x gt12
• Correct answer is F.

39
Alternate Interior Angles
40
Solution

• F is not correct because the lines are not equal.
• G is not correct because nothing is given to show
  they are congruent.
• H is correct. We know that the lines are
  parallel because the alternate interior angles
  are marked as congruent.
• J is not correct because the shapes are not the
  same size.
• K. is not correct because the shapes are not the
  same size.

41
Coordinate Geometry
42
Solution

• A quick sketch of the triangle shows that it is
  right triangle. That makes it easy to pick out
  base and height. The base is 5 units and the
  height is 8 units.
• Area of triangle .5 (base)(height)
• .5 5 8 20
• Correct answer is C.

43
Triangle properties
44
Solution

• All triangles have a sum of 180. Two of the
  angles are going to have the same measure.
• The only possible combination is 40 40 100
  180.
• Correct answer is H.

45
Properties of Triangles
46
Solution

• All triangles angles must sum to 180.
• For Triangle ABC (small), the two remaining
  angles, r and s, must total 140.
• For Triangle ADE (large), also contains angle A,
  so the two remaining angles, v and t, must total
  140.
• R S T V 140 140 280
• Correct answer is J.

47
Perimeter and Ratios
48
Solution

• Perimeter is adding up the three sides. One side
  is 16, the other are 2x and 3x
• 2x 3x 16 66 5x 1666 5x 50 x10
• Sides are 16, 2(10) or 20, and 3(10)30
• The longest side is 30, which is answer choice C.

49
Area of triangle
50
Solution

• When you are finding the area of any right
  triangle, use the two legs, 8 and 15. It is
  where the right angle forms.
• A .5 base height
• .5 8 15 60
• Correct answer is B.

51
Perimeter and Area of Rectangles
52
Solution

• Draw the rectangle. Label the width x and the
  length 3x. Since the perimeter is 160, add the
  four sides of x x 3x 3x 160. x20 so the
  width is 20 and the length is 60.
• To find the area, multiple length times width.
• A20(60) 1200
• Answer choice B is correct.

53
Similar Triangles/Proportions
54
Solution

• We have to look at the triangles as similar and
  set up a proportion. Put similar sides in the
  same position of the proportion.
• 2000/1200 2640x/x. Cross multiple and solve to
  get 2000x3168000 1200 x
• Subtract 1200X from both sides 800x 3168000. x
  3960
• Correct answer is H.

55
Perimeter
56
Solution D
57
Similar Triangles
58
Solution

• The perimeter of the triangle given is 21. The
  perimeter of the second triangle is 42.
  Therefore, the side lengths double to get the
  second triangle.
• Our longest side in the first triangle is 10, so
  the longest side in the second triangle is 20.
• Correct answer is G.

59
Midpoint and Area of Squares
60
Solution

• The midpoint means that each smaller segment on
  the outside square if 5. Since the two sides of
  the right triangle are both 5, it is a 45-45-90
  triangle, so the hypotenuse is 5v2
• The area of a square is found by multiplying two
  sides, each are 5v2 and 5v2 50
• Correct answer is B.

61
Area and Pythagorean Theorem
62
Solution

• To find the area of the triangle, you need to
  know two legs, use pythagorean theorem or the act
  program to find the missing leg of 6.
• The area of a triangle is .5 base height.
• A .5 6 8 24
• Correct answer is G.