3. The diagonals of a rectangle are 18 cm long and intersect at a 60 angle. Find the area of the rectangle.
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1 Geometry Chapter 11 Remaining Problems from the Textbook 1. Find the area of a square with diagonals of length d. 2. The lengths of the sides of three squares are s, s + 1, and s + 2. If their total area is 365 square centimeters, find their total perimeter. 3. The diagonals of a rectangle are 18 cm long and intersect at a 60 angle. Find the area of the rectangle.
2 4. Suppose you have 40 meters of fencing with which to make a rectangular pen for a dog. If one side of the rectangle is x meters long, express the area of the pen in terms of x. 5. A farmer has 100 meters of fencing with which to make a rectangular corral. A side of a barn will be used as one side of the corral. If the width of the corral is x, express the length and the area in terms of x. 6. Find the area. Give both an exact and an approximate answer. 7. Find the area. Give both an exact and an approximate answer.
3 8. Find the area. Give both an exact and an approximate answer. 9. Find the area of an isosceles triangle with a 32 vertex angle and a base of 8 centimeters. 10. FG is the altitude to the hypotenuse of DEF. Name three similar triangles and find their areas.
4 11. If the area of PQRS is 36 and T is a point on PQ, find the area of RST. 12. If BC = 16, h = 5 and AM is a median of ABC, find the areas of ABC and AMB. 13. (a) Find the ratio of the areas of the areas of QRT and QTS. (b) If the area of QRS is 240, find the length of the altitude from S to QR.
5 14. An isosceles triangle has sides that are 5 cm, 5 cm, and 8 cm long. Find its area and the lengths of the three altitudes. 15. (a) Find the area of the right triangle in terms of a and b. (b) Find the area of the right triangle in terms of c and h. (c) Solve for h in terms of the other variables. (d) A right triangle has legs 6 and 8. Find the lengths of the altitude and the median to the hypotenuse.
6 16. The area of a rhombus is 100. Find the length of the two diagonals is one is twice as long as the other. 17. The base of a triangle is 1 cm longer than its altitude. If the area of the triangle is 210 square centimeters, how long is the altitude? 18. Find the area of the trapezoid. Give both an exact and an approximate answer.
7 19. Find the area of the trapezoid. Give both an exact and an approximate answer. 20. Find the area of the trapezoid. Give both an exact and an approximate answer. 21. The legs of an isosceles trapezoid are 10 cm. The bases are 9 cm and 21 cm. Find the area of the trapezoid and the lengths of the diagonals.
8 22. An isosceles trapezoid has bases 12 and 28. The area is 300. Find the height and the perimeter. 23. ABCDEF is a regular hexagon with side 12. Find the areas of the three regions formed when diagonals AC and AD are drawn. 24. A trapezoid of area 100 square centimeters has bases of 5 centimeter and 15 centimeters. Find the areas of the two triangles formed by extending the legs until they intersect.
9 25. Find the exact area of each trapezoid. 26. Find the exact area of the trapezoid.
10 In problems #27-29, three regular polygons are inscribed in circles with radii 1. Find the apothem, the perimeter, and the area of each polygon
11 30. Find the perimeter and area of a regular dodecagon inscribed in a circle with a radius of A basketball rim has diameter 18 inches. Find the circumference of the rim and the area it encloses. 32. When a basketball player is shooting a free throw, the other players must stay out of the shaded region shown. This region consists of a semicircle together with a rectangle. Find the area of this region.
12 33. If 6 ounces of dough are needed to make an 8-inch pizza, how much dough will be needed to make a 16-inch pizza of the same thickness? (Hint: Compare the area of the pizza tops.) 34. One can of pumpkin pie mix will make a pie of diameter 8 inches. If two cans of pie mix are used to make a larger pie of the same thickness, find the diameter of that pie. 35. A school s wrestling mat is a square with 40-ft sides. A circle 28 feet in diameter is painted on the mat. No wrestling is allowed outside the circle. Find the area of the part of the mat that is not used for wrestling. 36. An advertisement states that a Roto-Sprinkler can water a circular region with area 1000 square feet. Find the diameter of this region to the nearest foot.
13 37. Which is the better buy, a 10-inch pizza costing $5 or a 15-inch pizza costing $9? 38. A Ferris wheel has diameter 42 feet. To the nearest foot, how far will a rider travel during a 4- minute ride if the wheel rotates once every 20 seconds? 39. A target consists of four concentric circles with radii 1, 2, 3, and 4. Find the area of the bull s eye and of each ring of the target.
14 40. The shaded region in the diagram is formed by drawing four quarter-circles within a square of side 8. Find the area of the shaded region. 41. The figure at the right consists of semicircles within a circle. Find the area of each shaded region. 42. Find the area of the shaded region.
15 43. Find the area of the shaded region. 44. Find the area. 45. Draw a square and its inscribed and circumscribed circles. Find the ratio of the areas of these two circles.
16 46. Chord AB is 18 cm long and the radius of the circle is 12 cm. Find the area of the shaded region to the nearest square centimeter. 47. The diagram shows some dimensions in a baseball stadium. H represents home plate. Find: (a) the area of the fair territory (the shaded region) (b) the area of the foul territory (the unshaded region)
17 48. ABCD is a square with sides 8 cm long. Two circles each with radius 8 cm are drawn, one with center A and the other with center C. Find the area of the region inside both circles. 49. Two circles have radii 6 cm and their centers are 6 cm apart. Find the area of the region common to both circles.
18 50. Three circles with radii 6 are tangent to each other. Find the area of the region enclosed between them.
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