1. Geometry and Measurement Circles, Lines, quadrilaterals and other polygons are done by: Ali Mohammed Ali Grade 12-04

2. Circles A circle is the locus of all points equidistant from a central point.

3. Definitions Related to Circles: arc: a curved line that is part of the circumference of a circle. circumference: the distance around the circle. diameter: the longest distance from one end of a circle to the other. pi : A number, 3.141592..., equal to (the circumference) / (the diameter) of any circle. radius: distance from center of circle to any point on it. tangent of circle: a line perpendicular to the radius that touches ONLY one point on the circle.

4. Related questions to Circles. Question #1 In a large field, a circle with an area of 144π square meters is drawn out. Starting at the center of the circle, a groundskeeper mows in a straight line to the circle's edge. He then turns and mows ¼ of the way around the circle before turning again and mowing another straight line back to the center. What is the length, in meters, of the path the groundskeeper mowed? a) 24 + 6π b) 12 + 6π c) 12 + 36π d) 24 + 36π e) 24π

5. Question #2 If the area of a circle is four times larger than the circumference of that same circle, what is the diameter of the circle? a) 4 b) 16 c) 32 d) 2 e) 8

6. Question # 3 A star is inscribed in a circle with a diameter of 30, given the area of the star is 345, find the area of the shaded region, rounded to one decimal. a) 351.5 b) 356.5 c) 341.5 d) 346.5 e) 361.5

7. Question #4 In the figure above that includes Circle O, the measure of angle BAC is equal to 35 degrees, the measure of angle FBD is equal to 40, and the measure of arc AD is twice the measure of arc AB. Which of the following is the measure of angle CEF? The figure is not necessarily drawn to scale, and the red numbers are used to mark the angles, not represent angle measures. a) 30 b) 75 c) 60 d) 110 e) 80

8. Question #5 A park wants to build a circular fountain with a walkway around it. The fountain will have a radius of 40 feet, and the walkway is to be 4 feet wide. If the walkway is to be poured at a depth of 1.5 feet, how many cubic feet of concrete must be mixed to make the walkway? a) None of the other answers b) 1296π cubic feet c) 504π cubic feet d) 336π cubic feet e) 1936π cubic feet

9. Answers Explanation for question #1 Circles have an area of πr 2, where r is the radius. If this circle has an area of 144π, then you can solve for the radius: πr 2 = 144π r 2 = 144 r =12 When the groundskeeper goes from the center of the circle to the edge, he's creating a radius, which is 12 meters. When he travels ¼ of the way around the circle, he's traveling ¼ of the circle's circumference. A circumference is 2πr. For this circle, that's 24π meters. One-fourth of that is 6π meters. Finally, when he goes back to the center, he's creating another radius, which is 12 meters. In all, that's 12 meters + 6π meters + 12 meters, for a total of 24 + 6π meters. Answer is A

10. Explanation for question #2 Set the area of the circle equal to four times the circumference πr 2 = 4(2πr). Cross out both π symbols and one r on each side leaves you with r = 4(2) so r = 8 and therefore d = 16. Answer is B Explanation for question #3 The area of the circle is (30/2) 2 *3.14 (π) = 706.5, since the shaded region is simply the area difference between the circle and the star, it’s 706.5-345 = 361.5 Answer is E

11. Explanation for question #4 AD + AB + CD + BC = 360 AD + AB + 80 + 70 = 360 AD + AB = 210 Because AD = 2AB, we can substitute 2AB for AD. 2AB + AB = 210 3AB = 210 AB = 70 This means the measure of arc AB is 70 degrees, and the measure of arc AD is 2(70) = 140 degrees. Now, we have all the information we need to find the measure of angle CEF, which is equal to half the difference between the measure of arcs AD and CD. CEF = (1/2)(140 - 80) = (1/2)(60) = 30. The answer is 30. Answer is A

12. We are searching for the surface area of the shaded region. We can multiply this by the depth (1.5 feet) to find the total volume of this area. The radius of the outer circle is 44 feet. Therefore its area is 442π = 1936π. The area of the inner circle is 402π = 1600π. Therefore the area of the shaded area is 1936π – 1600π = 336π. The volume is 1.5 times this, or 504π. Answer is C

13. Quadrilaterals And Other Polygons Quadrilateral just means "four sides". But the sides have to be straight. There are special types of quadrilateral:  The Rectangle it s a four-sided shape where every angle is a right angle (90°).  The Rhombus it is a four-sided shape where all sides have equal length.  The Square it has equal sides and every angle is a right angle (90°).  The Parallelogram it has opposite sides parallel and equal in length. Also opposite angles are equal.  The Trapezoid it has a pair of opposite sides parallel.

14. Questions related to Quadrilaterals And Other Polygons 1-Which quadrilateral does not have two sets of parallel sides? Which quadrilateral does not have two sets of parallel sides? a) Square b) Rhombus c) Trapezoid d) Parallelogram 2-identify the following: a) Rectangle b) Triangle c) Square d) Parallelogram

15. 3- Based on the knowledge you have of triangles, what do you know about a parallogram? Measure the angles of the parallelogram with a protractor, and add them up. What is the sum? 4- Based on the knowledge you have of triangles, what do you know about a trapezoid? Measure the angles of the trapezoid with a protractor, and add them up. What is the sum?

16. Answers 1-trapezoid 2- parallelogram 3- 2 opposite equal angles, parallel sides, 2 sets equal sides angle sum is 360 degrees. 4- 1 set parallel sides, 4 vertices, unequal sides total angles = 360 degrees

17. Parallel and Intersecting Lines When a line intersects (or crosses) a pair of parallel lines, there are some simple rules that can be used to calculate unknown angles.  a=b(and c=d, and e=f) these are called vertically opposite angles.  a=c(and b=d) these are called corresponding angles.  b=c these are called alternate angles.  a+e=180 degree, because adjacent angles on a straight line add up to 180 degree. these are called supplementary angles. Note also, that c+e=180degree( allied or supplementary angles)

18. Questions related to Parallel and Intersecting Lines 1) Which line below is parallel to y-2= 3 / 4 x? 2) assume line a and line b are parallel if angle x is three bigger than twice the square of four of angle, then what is angle y? 3) Two line are described by the equations: y=3x+5 and 5y-25=15x which of the following is true about the equation for these two lines? 4) A line passes through the points(-1,-2) and (1,2). Which of the following lines is parallel to this line?

19. Answers 1. y= 3 / 4 x-5 2. 7 3. They represent the same lines. 4. The line between the points (-2,0) and (0,4). 5. 4x – 3y=2, 6y= 8x+9

20. Thank you

21. Solid Geometry Made by: Ali Khalfan Ali Grade: 12-04 Teacher Name: Mr. Abdul Salam Abdulla Dulaimi

22. Information about:  Solid Geometry: In mathematics, [solid geometry] was the traditional name for the geometry of three- dimensional Euclidean space — for practical purposes the kind of space we live in. It was developed following the development of plane geometry. Stereometry deals with the measurements of volumes of various solid figures including cylinder, circular cone, truncated cone, sphere, and prisms. The Pythagoreans had dealt with the regular solids, but the pyramid, prism, cone and cylinder were not studied until the Platonists. Eudoxus established their measurement, proving the pyramid and cone to have one- third the volume of a prism and cylinder on the same base and of the same height, and was probably the discoverer of a proof that the volume of a sphere is proportional to the cube of its radius.

23. Some videos about [Solid Geometry]:

24. Some videos about [Solid Geometry]:

25. Some [Solid Geometry] Shapes :

26. Some Question about [Solid Geometry] : 1. A cubic box has sides of length x. Another cubic box has sides of length 4x. How many of the boxes with length x could fit in one of the larger boxes with side length 4x? A. 16 B. 40 C. 4 D. 80 E. 64

27. Some Question about [Solid Geometry] : 2. A cube weighs 5 pounds. How much will a different cube of the same material weigh if the sides are 3 times as long? A. 10 pounds B. 135 pounds C. 45 pounds D. 15 pounds E. 50 pounds

28. Some Question about [Solid Geometry]: 3. I have a hollow cube with 3” sides suspended inside a larger cube of 9” sides. If I fill the larger cube with water and the hollow cube remains empty yet suspended inside, what volume of water was used to fill the larger cube? A. 72 in 3 B. 216 in 3 C. 702 in 3 D. 73 in 3 E. 698 in 3

29. Some Question about [Solid Geometry]: 4. If the volume of a cube is 50 cubic feet, what is the volume when the sides double in length? A. 300 cu ft B. 200 cu ft C. 100 cu ft D. 500 cu ft E. 400 cu ft

30. Some Question about [Solid Geometry]: 5. A cube is inscribed in a sphere of radius 1 such that all 8 vertices of the cube are on the surface of the sphere. What is the length of the diagonal of the cube? A. √(2) B. 2 C. √(3) D. 8 E. 1

31. Explain The Answers: 1. [E], because the volume of a cubic box is given by (side length)3. Thus, the volume of the larger box is (4x)3 = 64x3, while the volume of the smaller box is x3. Divide the volume of the larger box by that of the smaller box, (64x3)/(x3) = 64. 2. [B], because cube that has three times as long sides is 3x3x3=27 times bigger than the original. Therefore, the answer is 5x27= 135.

32. Explain The Answers: 3. [C], because the volume of both cubes and then subtract the smaller from the larger. The large cube volume is 9” * 9” * 9” = 729 in 3 and the small cube is 3” * 3” * 3” = 27 in 3. The difference is 702 in 3. 4. [E], because using S as the side length in the original cube, the original is s*s*s. Doubling one side and tripling the other gives 2s*2s*2s for a new volume formula for 8s*s*s, showing that the new volume is 8x greater than the original.

33. Explain The Answers: 5. [B], because since the diagonal of the cube is a line segment that goes through the center of the cube (and also the circumscribed sphere), it is clear that the diagonal of the cube is also the diameter of the sphere. Since the radius = 1, the diameter = 2.

34. Thanks For Your Attention

35. Triangles Done by: Omar Ali Ahmed Ali Shaheen G12-04

36. What is SAT? The SAT and SAT Subject Tests are a suite of tools designed to assess your academic readiness for college. These exams provide a path to opportunities, financial support and scholarships, in a way that's fair to all students. The SAT and SAT Subject Tests keep pace with what colleges are looking for today, measuring the skills required for success in the 21st century.

37. Triangles A triangle has three sides and three angles. The three angles always add to 180°

38. Equilateral, Isosceles and Scalene There are three special names given to triangles that tell how many sides (or angles) are equal. There can be 3, 2 or no equal sides/angles:

39. Equilateral Triangle Three equal sides Three equal angles, always 60°

40. Isosceles Triangle Two equal sides Two equal angles

41. Scalene Triangle No equal sides No equal angles

42. What Type of Angle? Triangles can also have names that tell you what type of angle is inside.

43. Acute Triangle All angles are less than 90°

44. Right Triangle Has a right angle (90°)

45. Obtuse triangle Has an angle more than 90°

46. Combining the Names Sometimes a triangle will have two names, for example: Right Isosceles Triangle Has a right angle (90°), and also two equal angles Can you guess what the equal angles are?

47. Perimeter The perimeter is the distance around the edge of the triangle: just add up the three sides:

48. Area The area is half of the base times height. "b" is the distance along the base "h" is the height (measured at right angles to the base) Area = ½ × b × hThe formula works for all triangles.

49. Example Another way of writing the formula is bh/2 Example: What is the area of this triangle? Height = h = 12 Base = b = 20 Area = ½ × b × h = ½ × 20 × 12 = 120

50. The base Just be sure the "height" is measured at right angles to the "base”

51. Why is the Area "Half of bh"? Imagine you "doubled" the triangle (flip it around one of the upper edges) to make a square-like shape (it would be a "parallelogram" actually), THEN the whole area would be bh (that would be for both triangles, so just one is ½ × bh).

52. For instance

53. Question 1 2. A right circular cylinder has a radius of 3 and a height of 5. Which of the following dimensions of a rectangular solid will have a volume closest to the cylinder. (A) 4, 5, 5, (B) 4, 5, 6, (C) 5,5,5, (D) 5,5,6, (E) 5,6,6

54. Solution V = πr2h > V = π × 32 × 5 = 45π V = 45 × 3.142 = 141.39 We now have to test the volume of each of the rectangular to find out which is the closest to 141.39. (A) 4 × 5 × 5 = 100 (B) 4 × 5 × 6 = 120 (C) 5 × 5 × 5 = 125 (D) 5 × 5 × 6 = 150 (E) 5 × 6 × 6 = 180 Answer: (D) 5, 5, 6

55. Question 2 In the figure below, what is the value of y? (A) 40 (B) 50 (C) 60 (D) 100

56. Solution Step 1: Vertical angles being equal allows us to fill in two angles in the triangle that y° belongs to. Sum of angles in a triangle = 180°. So, y° + 40° + 80° = 180° > y° + 120° = 180° y° = 60°

57. Vidoes http://www.youtube.com/watch?v=xz6gBA0M9FY http://www.youtube.com/watch?v=KVFwRA7kcLY http://www.youtube.com/watch?v=XFh_JC7OSrg http://www.youtube.com/watch?v=nfMkORv6ybc