Geometry B Honors Chapter 10 Practice Test 1. Find the area of a square whose diagonal is 10. 2. Each rectangle garden below has an area of 100. w x 20 10 z 4 y 8 [a] Find the missing dimension of each garden. [b] What length of fencing is needed to surround each garden? 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 P N
D O C 5. If the area of rectangle RCTN is 6 times the area of AECT, find the coordinates of A. A N T (24, 8) C R (-­‐12, 0) E 6. Find the area of the obtuse triangle. 16 30o 7 7.
Find the area of the triangle. 8.
Find the area of the triangle. 9.
Find the area of the triangle. 60o 12 12 2 12 10. 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. 11. Lines CF and AB are parallel and 10 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 10 A 16 B 12. Find the area of an equilateral triangle with perimeter of 45 m. 13. Find the area of the shaded triangular region. 8 5 6 12 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) 22. The consecutive sides of an isosceles trapezoid are in the ratio of 2:5:10:5 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 18. 17. Find the area of the parallelogram to the nearest tenth. 21. The height of a trapezoid is 10, 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 40. 20. Given a trapezoid with bases 6 and 15 and height of 7, find the median and the area. 120o 6 14 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. 24. Find the area of a square if the radius of its inscribed circle is 9. 18. Find the area of the parallelogram to the nearest tenth. 135o 10 17 19. If the diagonals of a rhombus are 10 and 24, find the area and the perimeter of the rhombus. (hint: area of a rhombus can be found by using 1
, where d1 and d2 are the diagonal A= dd
2
1 2
lengths) 25. Find the area of a regular hexagon if the radius of its inscribed circle is 12. 26. A circle of radius 12 is circumscribed about each regular polygon below. Find the area of each regular polygon. a. b. c. 27. Find the area of the shaded region in this regular polygon. (hint: h = 2a, where a is the apothem) 6 h 28. Find the area of the shaded region in this regular polygon. 6 29. A square is formed by joining the midpoints of alternate sides of a regular octagon. A side of the octagon is 10. a. Find the area of the octagon b. Find the area of the shaded region 10 17 30. Find the ratio of the area of the shaded triangle to the area of the whole triangle. 2 7 31. Find the ratio of the area of the shaded triangle to the area of the whole triangle. 8 2 32. 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, s = a + b + c . 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 15 33. 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? 34. 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) 35. Find the area of a triangle with sides of 41, 41, and 18 without using the Hero’s Formula. 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? 36. Find the area of a parallelogram with sides of 6 and 7 and included angle 45o. 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) 38. The diagonal of a square is 26. Find the square’s area. 39. Find the diagonal of a square whose area is 18. 40. Which has a greater area, a circle with circumference of 100 or a square with a perimeter of 100? How much greater? 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? 44. 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? 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 600. 1
60° 2 8 46. 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. 47. Find the number of sides an equiangular polygon has if each of its exterior angles is a. 60o b. 40o c. 36o d. 2o 48. In the stop sign shown, is ΔNTE scalene, isosceles, equilateral, or undetermined? 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? 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. 51. The measures of three of the angles of a quadrilateral are 40, 70, and 130. What is the measures of the fourth angle? 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. 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? 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? 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? ***** ANSWERS ******* 1. Side length = 5
29. Area of octagon = 484, Area of square = 289, so area of shaded region is 484 – 289 = 195 sq units 2 , Area = 50 units2 30. Since scale factor is 2:9, the area ratio is 4:81 2. [a] w = 5, x = 10, z = 12.5, y = 25 31. Since scale factor is 1:5, the area ratio is 1:25 [b] w  50, x  40, z  41, z  58 32. [a] 6 [b] s = 10, A = 10√2 3. x = 3, so dimensions are 9m and 15m [c] s = 20, A = 60 [d] s = 20, A = 2√5 4. since it’s a square, AB=AD=BC=CD=8 [e] s = 9, A = 6√3 [f] s = 21, A = 84 5.
AM=MB=BN=NC=CO=OD=DP=PA=4 (midpoint) 33. Area is 80, so it will cost $1200 PM = 4 2, Area = 32units 2 , Perimeter = 16 2units 34. A = 36 sq units, so BD has to be 8 units long. Area1 = 288, Area2 = 48, EC = 8
So B is at (-­‐2,0) 35. A = 360 sq units so A is at (18, 8)
36. Height is 3√2, so Area is 21√2 sq units 6. Height = 8, Area = 28 sq units 37. Diagonal is 10 long, Area is 120 sq units 7. Height = 6, Base = 6√3, A = 18√3 sq units 38. Derivation of the rhombus formula  A = 1 d 2 8. Base = height = 12, so Area = 72 sq units 2
so Area is 338 sq units 9. 36√3 sq units 10. x = 4, so base = 12, altitude = 8 units 39. Diagonal is 4 2 units 11. Aaah! Trick question!  all the triangles have the 40. Circle, by 168 sq units same area b/c they all have the same base and the 41. Height is 4√3, Area is 48√3 same height 42. [a] 1800° [b] 360° 43. [a] 7 12. Side length = 15, Area is 225 3 ≈ 97.4units 2 [b] 18 [c] 10 [d] 5 44. 4 sides, quadrilateral 4
13. Biggest triangle minus the two smaller triangles and 45. 6 sides, hexagon square leaves us with Area of 33 sq units 46. [a] always [b] never 14. Height = 24, so Area = 168 sq units 47. [a] 6 15. Area = 180 sq units 48. Isosceles, b/c regular polygons have congruent 16. Height = base = 9√2, so Area is 81 sq units [b] 9 [c] 10 [d] 180 diagonals 17. Height = 3√3, so Area is 42√3 sq units 49. One exterior angle of a hexagon is 60°. 18. Height = 5√2, so Area is 85√2 sq units So the sum of the interior angles of the unknown 19. Perimeter =52 units, Area is 120 sq units polygon is 9(60) = 540. 20. Median is 10.5 units, Area is 73.5 sq units So using S=(n-­‐2)180  540 = (n-­‐2)180 21. Base is 21 units long solving for n  n=5 so it’s a pentagon 22. x = 2, so height is 6, making Area = 72 sq units 23. [a] s = 12 [b] a = 6√3 50. [a] always [b] never [c] 216√3 sq units [c] always [d] sometimes 24. side length = 18, so Area is 324 sq units 51. 120° 25. a = 12, s = 8√3, n = 6, so Area = 288√3 sq units 52. x = 30, largest angle is 90°, so answer is 45 26. [a] 108√3 sq units [b] 288 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 53. [a] 5580° [b] n = 33 [c] 216√3 sq units