A line segment joins the midpoints of two opposite sides

27. Find the perimeter of the cyclic quadrilateral shown. 39. A walk-up ramp moves horizontally 20 ft while rising 4 ft. Use a calculator to approximate its length to the nearest tenth of a foot.

40. a Argue that the midpoint of the hypotenuse of a right

triangle is equidistant from the three vertices of the triangle. Use the fact that the congruent diagonals of a rectangle bisect each other. Be sure to provide a drawing. b Use the relationship from part a to find CM, the length of the median to the hypotenuse of right , in which , , and .

41. Two sets of rails railroad

tracks are equally spaced intersect but not at right angles. Being as specific as possible, indicate what type of quadrilateral WXYZ is formed.

42. In square ABCD not shown, point E lies on side . If

and , find BE .

43. In square ABCD not shown, point E lies in the interior of

ABCD in such a way that is an equilateral triangle. Find . m ∠DEC 䉭ABE AE = 10 AB = 8 DC BC = 8 AC = 6 m ∠C = 90° 䉭ABC In Exercises 29 to 31, explain why each statement is true.

29. All angles of a rectangle are right angles. 30. All sides of a rhombus are congruent.

31. All sides of a square are congruent.

In Exercises 32 to 37, write a formal proof of each theorem. 32. The diagonals of a square are perpendicular. 33. A diagonal of a rhombus bisects two angles of the rhombus.

34. If the diagonals of a parallelogram are congruent, the

parallelogram is a rectangle.

35. If the diagonals of a parallelogram are perpendicular, the

parallelogram is a rhombus.

36. If the diagonals of a parallelogram are congruent and

perpendicular, the parallelogram is a square.

37. If the midpoints of the sides of a rectangle are joined in

order, the quadrilateral formed is a rhombus. In Exercises 38 and 39, you will need to use the square root function of your calculator.

38. A wall that is 12 ft long by 8 ft high has a triangular brace

along the diagonal. Use a calculator to approximate the length of the brace to the nearest tenth of a foot. 1

28. Find the perimeter of the square shown.

A B C D 39 52 25 3 冑 2 8 12 20 4 W X Y Z