3. The sides of a rectangle are in ratio fo 3:5 and the rectangle s area is 135m2. Find the dimensions of the rectangle.
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1 Geometry B Honors Chapter Practice Test 1. Find the area of a square whose diagonal is. 7. Find the area of the triangle. 60 o Each rectangle garden below has an area of Find the area of the triangle. w x 20 z 4 y 8 [a] Find the missing dimension of each garden. 9. Find the area of the triangle [b] What length of fencing is needed to surround each garden?. In a triangle, a base and its altitude are in ratio of 3:2. The triangle s area is 48. Find the base and the altitude. 3. The sides of a rectangle are in ratio fo 3:5 and the rectangle s area is 135m2. Find the dimensions of the rectangle. 4. The area of square ABCD is 64 units2. MNOP is formed by joining the midpoints of the sides of ABCD. Find the area and perimeter of MNOP. A M B 11. Lines CF and AB are parallel and mm apart. Several triangles with base AB and a vertex on CF have been drawn below. Which triangle has the largest area? Explain. C D E F P N A 16 B D O C 5. If the area of rectangle RCTN is 6 times the area of AECT, find the coordinates of A. N A T (24, 8) 12. Find the area of an equilateral triangle with perimeter of 45 m. R (- 12, 0) E C 6. Find the area of the obtuse triangle. 13. Find the area of the shaded triangular region o 6 12
2 14. Find the area of a triangle whose sides are 25, 25, and 14. Hint: altitude is perpendicular bisector to the base of an isosceles triangle) 20. Given a trapezoid with bases 6 and 15 and height of 7, find the median and the area. 21. The height of a trapezoid is, and the trapezoid s area is 130. If one base is 5, find the other base. 15. Find the area of a right triangle whose legs are 9 and The consecutive sides of an isosceles trapezoid are in the ratio of 2:5::5 and the trapezoid s perimeter is 44. Find the area of the trapezoid. (hint: first find the lengths of each side. Then use PT to find the height and then area) 16. Find the area of an isosceles right triangle with a hypotenuse of Find the area of the parallelogram to the nearest tenth. 23. The radius of a regular hexagon is 12. a. Find the length of one side. b. Find the length of the apothem. c. Find the area of the polygon o Find the area of the parallelogram to the nearest tenth. 24. Find the area of a square if the radius of its inscribed circle is o 25. Find the area of a regular hexagon if the radius of its inscribed circle is If the diagonals of a rhombus are and 24, find the area and the perimeter of the rhombus. (hint: area of a rhombus can be found by using, where d 1 and d 2 are the diagonal A = 1 2 d 1 d 2 lengths)
3 26. A circle of radius 12 is circumscribed about each regular polygon below. Find the area of each regular polygon. a. 30. Find the ratio of the area of the shaded triangle to the area of the whole triangle. 7 2 b. 31. Find the ratio of the area of the shaded triangle to the area of the whole triangle. c. 27. Find the area of the shaded region in this regular polygon. (hint: h = 2a, where a is the apothem) 28. Find the area of the shaded region in this regular polygon. h A useful formula for finding the area of a triangle was developed nearly 2000 years ago by the mathematician Hero of Alexandria. His formula is A = s(s a)(s b)(s c), where a, b, and c are side lengths and s is the semiperimeter, a + b + c s =. 2 Use Hero s Formula to find the area of the triangles with sides of the following lengths. a. 3, 4, and 5 b. 5, 6 and 9 c. 8, 15, and 17 d. 3, 3, and 4 e. 3, 7, and 8 f. 13, 14, and A rectangular driveway is to be paved. The driveway is 20 meters long and 4 meters wide. The cost will be $15 per square meter. What is the total cost of paving the driveway? 29. A square is formed by joining the midpoints of alternate sides of a regular octagon. A side of the octagon is. a. Find the area of the octagon b. Find the area of the shaded region Find the coordinates of B so that ΔABD will have the same area as ΔACD. A (0, 9) C (14, 0) B(?,?) D (6, 0)
4 35. Find the area of a triangle with sides of 41, 41, and 18 without using the Hero s Formula. 36. Find the area of a parallelogram with sides of 6 and 7 and included angle 45 o. 42. On a clock, a segment is drawn connecting the mark at the 12 and the mark at the 1. Then another segment is drawn connecting the mark at the 1 and the mark at the 2. Segments are drawn all the way around the clock in this same fashion. a. What is the sum of the measures of the angles of the polygon formed? b. What is the sum of the measures of the exterior angles, one per vertex, of the polygon? 37. Find the area of a rhombus whose perimeter is 52 and longer diagonal is 24. (Hint: draw a rhombus and its diagonals. Use PT to find length of half the other diagonal. Then use the formula from #19) 43. How many sides does a polygon have if the sum of the measures of its angles is a. 900? b. 2880? c. 1440? d. Six right angles? 38. The diagonal of a square is 26. Find the square s area. 39. Find the diagonal of a square whose area is In what polygon is the sum of the measures of the exterior angles, one per vertex, equal to the sum of the measures of the angles of the polygon? 40. Which has a greater area, a circle with circumference of 0 or a square with a perimeter of 0? How much greater? 45. In what polygon is the sum of the measures of the interior angles of the polygon equal to twice the sum of the measures of the exterior angles, one per vertex? 41. Find the area of a parallelogram with sides 12 and 8 and included angle of Tell whether each statement is true ALWAYS, SOMETIMES, or NEVER. a. As the number of sides of a polygon increases, the number of exterior angles increases. b. As the number of sides of a polygon increases, the sum of the measures of the exterior angles increases.
5 47. Find the number of sides an equiangular polygon has if each of its exterior angles is a. 60 o b. 40 o c. 36 o d. 2 o 52. The measures of the angles of a triangle are in the ratio of 1:2:3. Find half the measure of the largest angle. 48. In the stop sign shown, is ΔNTE scalene, isosceles, equilateral, or undetermined? 53. If a polygon has 33 sides, what is a. The sum of the measures of the angles of the polygon? b. The sum of the measures of the exterior angles, one per vertex, of the polygon? 49. The sum of a polygon s angle measures is nine times the measure of an exterior angle of a regular hexagon. What is the polygon s name? EXTRA CREDIT: What is the name of an equiangular polygon if the ratio of the measure of an interior angle to the measure of an exterior angle is 7:2? 50. Tell whether each statement is true Always, Sometimes, or Never a. If the number of sides of an equiangular polygon is doubled, the measure of each exterior angle is halved. b. The measure of an exterior angle of a decagon is greater than the measure of an exterior angle of a quadrilateral. c. A regular polygon is equilateral. d. An equilateral polygon is regular. EXTRA CREDIT: If the sum of the measures of the angles of a polygon is increased by 900, how many sides will have been added to the polygon? 51. The measures of three of the angles of a quadrilateral are 40, 70, and 130. What is the measures of the fourth angle?
6 ***** ANSWERS ******* 1. Side length = 5 2, Area = 50 units 2 2. [a] w = 5, x =, z = 12.5, y = 25 [b] w 50, x 40, z 41, z x = 3, so dimensions are 9m and 15m 4. since it s a square, AB=AD=BC=CD=8 AM=MB=BN=NC=CO=OD=DP=PA=4 (midpoint) PM = 4 2, Area = 32units 2, Perimeter = 16 2units 5. Area = 288, Area 1 2 = 48, EC = 8 so A is at (18, 8) 6. Height = 8, Area = 28 sq units 7. Height = 6, Base = 6 3, A = 18 3 sq units 8. Base = height = 12, so Area = 72 sq units sq units. x = 4, so base = 12, altitude = 8 units 11. Aaah! Trick question! all the triangles have the same area b/c they all have the same base and the same height 12. Side length = 15, Area is units Biggest triangle minus the two smaller triangles and square leaves us with Area of 33 sq units 14. Height = 24, so Area = 168 sq units 15. Area = 180 sq units 16. Height = base = 9 2, so Area is 81 sq units 17. Height = 3 3, so Area is 42 3 sq units 18. Height = 5 2, so Area is 85 2 sq units 19. Perimeter =52 units, Area is 120 sq units 20. Median is.5 units, Area is 73.5 sq units 21. Base is 21 units long 22. x = 2, so height is 6, making Area = 72 sq units 23. [a] s = 12 [b] a = 6 3 [c] sq units 24. side length = 18, so Area is 324 sq units 25. a = 12, s = 8 3, n = 6, so Area = sq units 26. [a] 8 3 sq units [b] 288 sq units [c] sq units 27. hexagon a = 3 3, s = 6, n = 6 rectangle b = 6, h = 6 3, so Area = 36 3 sq units 28. rectangle 29. Area of octagon = 484, Area of square = 289, so area of shaded region is = 195 sq units 30. Since scale factor is 2:9, the area ratio is 4: Since scale factor is 1:5, the area ratio is 1: [a] 6 [b] s =, A = 2 [c] s = 20, A = 60 [d] s = 20, A = 2 5 [e] s = 9, A = 6 3 [f] s = 21, A = Area is 80, so it will cost $ A = 36 sq units, so BD has to be 8 units long. So B is at (- 2,0) 35. A = 360 sq units 36. Height is 3 2, so Area is 21 2 sq units 37. Diagonal is long, Area is 120 sq units 38. Derivation of the rhombus formula A = 1 2 d 2 so Area is 338 sq units 39. Diagonal is 4 2 units 40. Circle, by 168 sq units 41. Height is 4 3, Area is [a] 1800 [b] [a] 7 [b] 18 [c] [d] sides, quadrilateral sides, hexagon 46. [a] always [b] never 47. [a] 6 [b] 9 [c] [d] Isosceles, b/c regular polygons have congruent diagonals 49. One exterior angle of a hexagon is 60. So the sum of the interior angles of the unknown polygon is 9(60) = 540. So using S=(n- 2) = (n- 2)180 solving for n n=5 so it s a pentagon 50. [a] always [b] never [c] always [d] sometimes x = 30, largest angle is 90, so answer is [a] 5580 [b] n = 33
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