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 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.

Circumference

Solution

Circumference of a circle is either C=2*pi*radius or C=pi*diameter. You choose the formula which helps you answer the question. Here, we are looking for diameter, so I am using the second formula. Since the circumference is 16*pi, I know from the formula that the diameter is 16, so the correct answer is D.

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.