Download MCHS ACT Review

MCHS ACT Review
Plane Geometry
Created by Pam Callahan
 Spring 2013 Edition
Angle Measures
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.
Area of Rectangles & Triangles
Solution:
 Square feet means that we are looking for area. The
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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.
Area and diameter
Solution
 Area of a circle is A=pi*radius squared. Here,
we are looking for diameter, so I am using the
second formula. Since the area is 16*pi, I
know from the formula that the radius is 4 and
the diameter is 8, so the correct answer is C.
Arcs and Circles
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.
Geometric Probability
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 don’t 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 =
pi*radius2= 100*pi. The area of the circle
which contains 30, 40, and 50 has a radius of
6 and an area of 36*pi. Divide 36*pi/100*pi,
the pi’s cancel out, and the area is .36 or
36%. The correct answer is A.
45-45-90 Special Right Triangle
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-√2. Since the sides are 7, the
hypotenuse must be 7√2, which is answer
choice G.
Volume of Cubes
Solution:
 Volumes of cubes are sides3. The original
cube has sides of x, so the volume is x*x*x or
x3. The edges of the large cube is 2x, so its
volume is 2x*2x*2x or 8x3. Divide the two
volumes (8x3/x3) is 8 times. The correct
answer is D.
Isosceles Triangles
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.
Area of Circles
Solution:
 The area of a circle is A=pi*radius2. 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: pi*92 = 254.47: Small: pi*4.52 = 63.62
 Difference of 190.85, which is answer choice
D.
Complementary Angles
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.
Circles
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.
Volume of Pyramid
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.
Quadrilaterals
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.
Area of Rectangles
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.
Volume of Rectangular Prism
Solution:
 The volume of any rectangular prism is length
* width * height.
 Multiply 3 * 2 * 2 = 12
 Correct answer is G.
Area of Trapezoid
Solution:
 The formula for the area of a trapezoid is:
A = .5 * height (b1 + b2)
 A = .5 (4)(7 + 5) = 24
 Correct answer is G.
Pythagorean Theorem
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.
Triangle Inequality
Solution:
 The rule is that the sum of any two sides must
be greater than the third side, or you have a
gap and can’t construct the triangle.
 Let the third side be x and look at three
inequalities
 X + 4.5 > 7.5 so x > 3.5
 X + 7.5 > 4.5 so x > -3.5 (can’t use this one)
 7.5 + 4.5 > x so x >12
 Correct answer is F.
Alternate Interior Angles
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.
Coordinate Geometry
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.
Triangle properties
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.
Properties of Triangles
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.
Perimeter and Ratios
Solution:
 Perimeter is adding up the three sides. One
side is 16, the other are 2x and 3x
 2x + 3x + 16 = 66; 5x + 16=66; 5x = 50; x=10
 Sides are 16, 2(10) or 20, and 3(10)=30
 The longest side is 30, which is answer
choice C.
Area of triangle
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.
Perimeter and Area of Rectangles
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. x=20
so the width is 20 and the length is 60.
 To find the area, multiple length times width.
 A=20(60) = 1200
 Answer choice B is correct.
Similar Triangles/Proportions
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 = 2640+x/x. Cross multiple and
solve to get 2000x=3168000 + 1200 x
 Subtract 1200X from both sides 800x =
3168000. x = 3960
 Correct answer is H.
Perimeter
Solution: D
Similar Triangles
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.
Midpoint and Area of Squares
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 5√2
 The area of a square is found by multiplying
two sides, each are 5√2 and 5√2 = 50
 Correct answer is B.
Area and Pythagorean Theorem
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.