Chapter 1.2, Problem 8AEXP

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.

A quadrilateral RSTV is circumscribed about a circle so that its tangent sides are at the endpoints of two intersecting diameters. aWhat types of quadrilateral is RSTV? bIf the diameters are also perpendicular, what type of quadrilateral is RSTV?

Draw an obtuse triangle and construct the three perpendicular bisectors of its sides. Do the perpendicular bisectors of the three sides appear to meet at a common point?

What is the general name of the point of concurrence for the three altitudes of a triangle?

In the illustration, point M is the midpoint of the hypotenuse of right triangle AOB. Show that the area of rectangle OLMN is one-half of the area of triangle AOB.

A trapezoid is a four-sided figure that contains one pair of parallel sides. Which sides of the trapezoid MNPQ appear to be parallel?

In quadrilateral RSTV, the midpoints of consecutive sides are joined in order. Try drawing other quadrilaterals and joining their midpoints. What can you conclude about the resulting quadrilateral in each case?

a Is it really necessary to construct all three bisectors of the angles of a triangle to locate its incenter? b Is it really necessary to construct all three perpendicular bisectors of the sides of a triangle to locate its circumcenter?

Generalize your findings in Exercise 33. 33. Can you use the construction for the midpoint of a segment to divide a line segment into a three congruent parts? c six congruent parts? b four congruent parts? d eight congruent parts?