Simplifying square roots Sometimes we need to leave the answer with a radical or square root sign. When we have to do it that way, the calculator isn't very helpful. We need to know how to simplify a square root. This is possible because we can re-write the number as a product of factors and take the square root of the parts that are perfect squares and leave the rest in a square root sign. For example, if I need to take the square root of 12, and I use a calculator I get an answer that looks something like this 3.464101615. If I'm asked to give the answer in terms of square roots, then the calculator answer won't help me. I need to try a different approach. I know that 12 isn't a perfect square so it won't have a simple integer answer, so I look for factors of 12 that are perfect squares. 4 is a factor of 12 and 4 is a perfect square so I will re-write the square root like this 12 = 4 x 3 = 4 3 = 2 3. I can't break it down any farther, so I write it like this 2 3 which means 2 times the square root of 3. If you multiply that on your calculator you will get 3.464101615. They are the same number, they just look different. One advantage to writing answers as square roots is that they are exact, whereas long decimal answers are usually rounded and approximated. 1. Simplify 32 2. Simplify 75 3. Simplify 8 4. Simplify 200 5. In the triangle below angle c is a right angle, and sides AC is equal to side BC. If side AC = 2, what is the length of side AB? Give the answer in terms of a simplified square root. 6. How many units long is one of the sides of a square that has a diagonal 20 units in length? A. 10 B. 10 2 C. 15 D. 20 E. 15 2 1
Exterior angles of a triangle An exterior angle of a triangle is equal to the sum of the remote interior angles - in other words, the two interior angles on the opposite side of the triangle. In the figure show below, the measure of angle x is 150 since the sum of the remote interior angles is 50 + 100 7. Find the measure of angle x in the figure to the right. 8. What is the value of x in the figure below? The three exterior angles of any triangle add up to 360. In the figure below a + b + c = 360. 2
9. In the figure below angle a is 120 and angle c is 90. What is the measure of angle b? Similar triangles Similar triangles have the same shape but are different sizes. Corresponding angles are equal and corresponding sides are proportional. In the figure below, triangle ABC is similar to triangle DEF. Similar triangles may be facing different directions which might make it more difficult to see that they are the same shape, such as the figures below. These triangles are similar because they have the same angles. That means that the corresponding sides are proportional. We can use this information to find missing values. To find the missing value of s in the figure above, set up a proportion. Since side bc of the first triangle is proportional to side bc of the second triangle and side ab of the first triangle is proportional to side ab of the second triangle then 3 4 = 6 s. Solve for s by cross multiplying. 4 x 6 = 3s, therefore s = 8 3
10. In the figure at right, angle ABC is a right angle and DF is parallel to AC. If AB is 12 inches long, BC is 9 inches long. AD is 6 inches long. What is the area, in square inches, of triangle DBF? 11. The Similar Triangles movie theater company use the Illuminator 100 light bulb in their projectors, but the new manager wants to switch to the Megabulb 100X, a more powerful light bulb that projects movies onto larger screens farther away. The Megabulb 100X projects movies onto screens 108 feet wide and 180 feet from the projector. The original Illuminator 100 projects movies only 81 feet wide. How much farther from the projector, in feet, is the screen for the Megabulb 100X than the screen for the Illuminator 100? 12. A person 5 feet tall casts a shadow 8 feet long. At the same time, a nearby tree casts a shadow 24 feet long. What is the height of the tree? 4
13. In the figure at right, angle M is congruent to angle K, angle N and angle L are right angles. Solve for u. 14. In the figure at right, QS is parallel to PT. The length of RS is 2 and the length of ST is 6. The perimeter of PRT is 24 inches. What is the perimeter of QRS? Hint: When parallel lines make a big triangle and a little triangle as they do here, the triangles are similar because they have the same angle measurements. The length of RS is 2 and it is a side of the small triangle and the length of RT is the corresponding side of the large triangle. Side RT is 8 (RS + ST). With this information we can make a ratio. RS Perimeter of triangle QRS = RT Perimeter of triangle PRT 15. For the triangles in the figure below, which of the following ratios of side lengths is equivalent to the ratio of the perimeter of ABC to the perimeter of DAB? A. AB:AD B. AB:BD C. AD:BD D. BC:AD E. BC: BD 5
16. Line segments GH, JK and LM are parallel and intersect line segments FL and FM as shown in the figure below. The ratio of the perimeter of FJK to the perimeter of FLM is 3:5, and the ratio of FH to FM is 1:5. What is the ratio of GJ to FG? A. 1:5 B. 1:3 C. 1:2 D. 2:1 E. 5:3 Hint: Because all triangles share angle F and the lines are parallel, the three triangles are similar, which means all their sides are proportional and therefore the perimeters are also proportional. The ratio of the perimeter of triangle FJK to triangle FLM is 3:5 and the ratio of FH to FM is 1:5. From the proportions given, you can determine that the ratio of FGH to FJK to FLM is 1:3:5. Therefore the ratio of FG to FJ to FL is also 1:2:3. Try substituting values in for the sides and see what works. Remember to get the ratio in the right order. 6
Isosceles triangles Isosceles triangles have two equal sides and two equal angles. When an image is inscribed in a circle we can often create isosceles triangles to help solve for unknown information. For example, in the image below, O is the center of the circle, and C, D, and E are points on the circumference of the circle. If angle OCD measures 70 and angle OED measures 45, what is the measure of angle COE? At first glance, it looks like we don't have enough information to answer the question. But we can add another line from point O to point D and that changes everything. Draw a line from point O to point D. Because OC is a radius of the circle and OD is also a radius of the circle they are the same length and that means that triangle OCD is an isosceles triangle and angle ODC is also 70. Now for triangle ODE. Since OD and OE are both radii, they are the same length and triangle ODE is an isosceles triangle. That means that angle EDO is 45. To find angle COE, simply add angle COD and angle DOE. Therefore, angle COE is 130. 17. In the figure below, O is the center of the circle, and C, D, and E are points on the circumference of the circle. If angle OCD measures 65 and angle OED measures 35, what is the measure of angle CDE? 7
18. In the following circle, chord AB passes through the center of circle O. If radius OC is perpendicular to chord AB and has a length of 12 centimeters, what is the length of chord BC, to the nearest tenth of a centimeter? 19. Points G and H lie on circle F as shown below. If the measure of angle FGH is 30, then what is the measure of central angle GFH? 20. In the figure below, F, G, H, and J are collinear. FG, GK, and HK are line segments of equivalent length, and the measure of angle JHK is 110. What is the degree measure of angle GFK? 8
Angle inscribed in a semicircle The angle inscribed in a semicircle is always a right angle (90 ) no matter where the third point lies on the circle. In the circle below, A, O, and B are collinear points on the diameter of circle O. Point P lies on the circle. Angle P must be a right angle. 21. In the figure below, BC is a diameter of the circle. What is the measure of angle A? 22. In the figure above, if C = 20, what is the measure of angle B? 23. Point O is the center of the circle shown at right, and XZ is the diameter of the circle. If XZ = 12 inches, Y lies on the circle, and OX = XY, then what is the area, in square inches, of XYZ? 9
Special right triangles. When using the Pythagorean theorem we often get answers with square roots or long decimals. There are a few special right triangles that give integer answers. The most often seen is the 3-4-5 right triangle. If one leg is 3 and the other is 4 then the hypotenuse will be 5. When you are able to recognize this kind of triangle, you don't even have to use the Pythagorean theorem, which makes things simpler and easier. Also, try to recognize multiples of the 3-4-5 right triangle. For example, all the triangles below are variations of the 3-4-5 right triangle. Four times a 3-4-5 right triangle makes a 12-16-20 triangle. Ten times a 3-4-5 makes a 30-40-50 right triangle. One half times a 3-4-5 right triangle makes a 1.5-2-2.5 triangle. 24. In the right triangle below, one leg is 30 and the hypotenuse is 50. Find b. 10
25. As shown in the figure below, triangle XYZ is a right triangle. Quadrilaterals ABYX, CDZY, and EFXZ are squares. If the measure of side x = 6 inches, and the measure of hypotenuse z = 10 inches, what is the area of quadrilateral EFXZ? Special right triangles - angles We've discussed special right triangles that can be recognized by their sides, there are also special right triangles that can be recognized by the angles. 45-45 -90 The first is an isosceles right triangle. We know that isosceles triangles have two sides the same length, that means that the angles opposite from those sides are also equal. Therefore every isosceles right triangle has the angles 45-45 -90. The sides of a 45-45 -90 triangle are in a ratio of 1:1: 2. In the example below, we have a 45-45 -90 triangle. The length of one side is 3. From that we can calculate all the other sides. Since one leg is 3, then the other leg is also 3. To find the hypotenuse multiply the length of the side by 2. So in this case it is 3 2 or about 4.24264... 11
26. Find the length of AT in the triangle below. 27. Solve for x in the triangle below. 28. If the length of side AC is 6, what is the length of side BC? If you know the length of the hypotenuse instead of one of the sides, it's not quite at simple, but still doable. Rather than multiply by 2 we will divide by 2. x 2 However, we usually don't leave a radical sign as a denominator. In order to get rid of the radical sign in the denominator we multiply both the numerator and the denominator by 2. x x 2 = x 2 2 2 2 Like this: 12
So in other words, if you know the length of the hypotenuse of a 45-45 -90 triangle, you can find the length of the side by multiplying by 2. For example. In the 45-45 -90 triangle below the hypotenuse is 10. To 2 find the length of one of the sides, multiply 10 X 2 = 5 2. Each side is 5 2 or about 7.07106... 2 29. In the 45-45 -90 triangle below the hypotenuse is 20. What is the length of one of the sides? 30. In the isosceles right triangle to the right, what is the length of the base leg? 31. A diagonal of a square creates two 45-45-90 triangles. What is the perimeter of square ABCD below? 13
There is another special right triangle with angles 30-60 -90. The sides of this triangle are in a ratio of 1: 3 : 2 In the triangle below. The length of the short leg is 5. To find the length of the longer leg multiply by 3. So it is 5 3 (or about 8.66025...). To find the length of the hypotenuse, simply multiply the short leg by 2. So the length of the hypotenuse is 10. 32. Find x and y in the triangle below. In the triangle below, the length of the hypotenuse is given. From that we can easily calculate the length of the short leg by dividing by 2. So side IR = 7in. To find the length of the longer leg, multiply the length of the shorter leg by 3. So side IT = 7 3 (or about 12.124355...). 14
33. Find j and k in the triangle below. If the length of the longer leg is given, it's a little more complicated, but not bad. In the figure below, the length of the longer side is 5. We know that the longer side is the shorter side multiplied by 3, so to find the shorter side we divide the longer side by 3. Step 1: 5 = x 3 Step 2: 5 = x (divide both sides by 3) 3 Step 3: 5 3 3 3 Step 4: 5 3 3 = x (we never leave a square root sign in the denominator, so, multiply top and bottom by 3. = x (multiply it out and simplify if possible) So the length of the shorter side is 5 3 3 hypotenuse, simply multiply the shorter side by 2. 10 3 3 (or about 2.88675...), from there it is easy to find the length of the 34. Find x and y in the triangle below. 15
35. What is the perimeter of a 30-60 -90 triangle with a short leg of 6 inches? 36. What is the perimeter of a 30-60 -90 triangle with a hypotenuse leg of 24 inches? 37. What is the perimeter of a 30-60 -90 triangle with a long leg of 5 inches? 38. Cindy wanted to stay off the main road for most of her morning jog. She started on Market Street ran 400 meters along a side road that runs at a 60 angle relative to Market Street. Then the road makes a slight turn so it's at an angle of 45 relative to Market Street and she jogged another 600 meters. At that point the side street joins up with Broad Street and she continued another 35 meters. How far north did she travel during her morning jog? A. 35 B. 115 C. 195 D. 50 2 + 30 3 E. 35 + 300 2 + 200 3 16
Triangle inequality theorem: The sum of the lengths of any two sides of a triangle must be greater than the third side. If these inequalities are NOT true, you do not have a triangle. In the example below c < a + b, b < a + c, a < b + c. That means that if a is 4 and b is 3 then side c has to be less than 7. If you think about it, it makes sense because a and b would have to be laid out flat on a straight line to make a 7 (3 + 4 = 7) and it wouldn't be a triangle any more, it would just be a straight line. And the third side couldn't be any longer than that, 8 for example, because 3 + 4 couldn't make an 8 even if they were laid out flat. 39. One side of a triangle is 3 inches long and another side is 6 inches long. Which of the following could NOT be the length of the other side? A. 10 inches B. 4 inches C. 5 inches D. 6 inches 40. In triangle ABC, side AB is 11 inches long and side BC is 50 inches long. Which of the following CANNOT be the length, in inches of side AC? A. 20 B. 60 C. 40 D. 50 41. If the lengths of all three sides of a triangle are integers, and one side is 9 inches long, what is the smallest possible perimeter of the triangle? Maximum area of a rectangle with a fixed perimeter. It's always a square. For example, if we were asked to find the largest area of a rectangle with a perimeter of 28 feet, we could use this information to find the answer. Since the largest area of a rectangle with a fixed perimeter is a square, we know that we have a square with a perimeter of 28 feet. That means that the square has sides that are 7 feet long. The area would be 7 x 7 = 49 square feet. 42. What is the maximum possible area, in square inches, of a rectangle with a perimeter of 8 inches? 17
Angles and sides of triangles. There is a relationship between an angle and the side opposite that angle. The larger the angle, the larger the side will be. In the triangle below, side a is opposite angle A, side b is opposite angle B, and side c is opposite angle C. In this figure, angle B is the smallest angle, therefore side b is the shortest side. Angle A is the next smallest angle, so side a is next in length. Angle C is the largest angle, therefore side c is the longest side. 43. In the triangle below, list the sides in order from least to greatest. Surface area of solids Surface area of a rectangular solid The surface of a rectangular solid - a box, or a cube, for example - is simply adding the area of each of the faces. If the length is l, the width is w, and the height is h, the formula is: Surface area = 2lw + 2wh + 2lh 44. Find the surface area of the rectangular solid to the right. 18
45. What is the surface area of a rectangular solid with a length of 8 inches, width of 6 inches, and height of 3 inches? 46. The surface area of the rectangular prism below is 280 square inches. The width (side b) is 6 inches and the height (h) is 5 inches. What is the length of the rectangular prism? Volume of solids Volume of a cylinder V=πr 2 h (in other words, find the area of the circle and multiply it by the height) 47. The radius of the cylinder at right is 3 inches and the height is 4 inches. What is the volume of the cylinder? 48. The volume of the right cylinder at right is 100π, if the height is 4 what is the radius? 19
The youth center has installed a swimming pool on level ground. The pool is a right circular cylinder with a diameter of 24 feet and a height of 6 feet. A diagram of the pool and its entry ladder is shown below. 49. To the nearest cubic foot, what is the volume of water that will be in the pool when it is filled with water to a depth of 4 feet? (note: The volume of a cylinder is given by πr 2 h where r is the radius and h is the height) Volume of a cone = 1 3 πr2 h 50. What is the volume of the cone below? Volume of a sphere = 4 3 πr3 51. What is the volume of a sphere with a radius of 3? 20
Questions that review parallel lines cut by a transversal 52. In the figure below, lines l and m are parallel, which of the following choices list a pair of angles that must be congruent? A. angle 1 and angle 2 B. angle 1 and angle 3 C. angle 2 and angle 3 D. angle 2 and angle 5 E. angle 3 and angle 5 53. In the figure below, line l is parallel to line m. Transversals t and u intersect at point A on l and intersect m at points C and B, respectively. Point X is on m, the measure of angle ACX is 130, and the measure of angle BAC is 80. How many of the angles formed by rays of l, m, t, and u have measure 50? 54. In the figure below, the three horizontal lines are parallel. The length of BC is 3, CD is 5, PQ is 4. What is the length of segment QR? 21
Questions that review triangles (sum of interior angles = 180 ) 55. In triangle ABC the measure of angle A is exactly 37, and the measure of angle B is less than or equal to 63. Which of the following phrases best describes the measure of angle C? A. Exactly 120 B. Exactly 100 C. Exactly 80 D. Greater than or equal to 80 E. Less than or equal to 80 Questions that review Pythagorean theorem 56. What is the height of the trapezoid below? Questions that review area/circumference of a circle 57. A circular pool is being build on a fenced rectangular lot 50 feet wide and 75 feet long. If a border of 10 feet is needed between the outside edge of the pool and the fence, what is the largest diameter of a pool that can be built? 58. For a project in Home Economics class, Sarah is making a tablecloth for a circular table 3 feet in diameter. The finished tablecloth needs to hang down 11 inches over the edge of the table all the way around. To finish the edge of the tablecloth, Sarah will fold under and sew down 1 inch of the material all around the edge. Sarah is going to use a single piece of rectangular fabric that is 60 inches wide. What is the shortest length of fabric, in inches, Sarah could use to make the tablecloth without putting any separate pieces of fabric together? 22
Questions that review area 59. An isosceles trapezoid has bases of length 4 inches and 8 inches. The area of the trapezoid is 30 square inches. What is the height of the trapezoid, in inches? (The area of a trapezoid is A = 1 h(b 2 1 + b 2 ). 60. Hexagon ABCDEF shown below was drawn on a grid with unit squares. Each vertex is at the intersection of 2 grid lines. What is the area of the hexagon, in square units? 61. What is the area in square units of the following figure? 23
Answers: 1. 4 2 2. 5 3 3. 2 2 4. 10 2 5. 2 2 6. B 7. 100 8. 20 9. 150 10. 13.5 in 2 11. 45 feet 12. 15 feet 13. 6 14. 6 15. A 16. D 17. 100 18. 6 2 or 8.5 cm 19. 120 20. 35 21. 90 22. 70 23. 18 3 or 31.18 in 2 24. 40 25. 64 in 2 26. 8 2 27. 2 2 28. 6 2 29. 10 2 30. 4 31. 60 2 or 84.85 32. x = 12; y = 6 3 33. j = 12 3; k = 12 34. x = 4 3; y = 8 3 35. 18 + 6 3 or 28.39 inches 36. 36 + 12 3 or 56.78 inches 37. 5 + 5 3 or 13.66 inches 38. E 39. A 40. A 41. 19 42. 4 in 2 43. BC, AB, AC 44. 122 45. 180 in 2 46. 10 in 47. 36π or 113.04 48. 5 49. 576π or 1,808.64 50. 18π or 56.52 51. 36π or 113.04 52. E 53. 8 54. 20 or 6.67 3 55. D 56. 3 57. 30 feet 58. 60 inches 59. 5 60. 20 61. 104 units 2 24