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MCHS ACT Review

MCHS ACT Review. Plane Geometry. Created by Pam Callahan. Spring 2013 Edition. Angle Measures. Solution:.

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MCHS ACT Review

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  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. Circumference

  8. 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.

  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 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.

  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-√2. Since the sides are 7, the hypotenuse must be 7√2, 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 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.

  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 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.

  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 is:A = .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 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.

  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 + 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.

  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.

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