![]() In these figures, the other two sides are parallel, too and so they meet not only the requirements for being a trapezoid (quadrilateral with at least one pair of parallel sides) but also the requirements for being a parallelogram. In fact, by the definition, even this is a trapezoid because it has “at least one pair of parallel sides” (and no other features matter), as are. The parallel sides may be vertical, horizontal, or slanting. ![]() (In English-speaking countries outside of North America, the equivalent term is trapezium.) Show that given ABC, at most two points D and D' are possible as the fourth vertex.A trapezoid is a quadrilateral with at least one pair of parallel sides. Proof: Start by constructing perpendiculars BF and AG as in this figure. ABCD is an isosceles trapezoid if and only if the base angles DAB and CBA are equal. Given a trapezoid ABCD with parallel sides AB and CD. Notice that if ABCD is a parallelogram, it is a (non-strict) trapezoid with BC = DA. If M is the midpoint of BC and N is the midpoint of CD, then the line MN is parallel to AB and CD.Ī trapezoid ABCD with parallel sides AB and CD is called an isosceles trapezoid if it is a strict trapezoid with BC = DA. If e is the line through E parallel to AB, then if e intersects BC in F and DA in G, the ratios BF/CF = AG/DG = AB/CD. Given a trapezoid ABCD with parallel sides AB and CD, with E the point of intersection of the diagonals AC and BD. The midpoints of AB and CD are collinear with the point of intersection of the diagonals. If k is any line through E intersecting AB in P and CD in Q, then AP/BP = CQ/DQ. Here are some other theorems about ratios and trapezoids that will be discussed in class. Here is an example constructing a point P that divides MN into segments of length 2/9 and 7/9, thus in the ratio 2/7. Or if a and b are integers, mark of a equal distances to make MM' and b of the same unit of distance to make NN'. For example, if a and b are lengths that have been constructed, then construct M' and N' with MM' = a and NN' = b. Construct a point M' on m and a point N' on n, with M' and N' on opposite sides of line MN, so that MM'/NN/ = a/b. If a and b are positive numbers and MN is a segment, to construct a point P so that MP/NP = a/b, construct two lines m and n perpendicular to MN, one through M and one through N. Since the triangles ABE and CDE are similar, then these are ratios of corresponding sides. If we label as r the ratio r = AB/CD, then the diagonals are divided by this ratio AE/CE = BE/DE = r. Given a trapezoid ABCD with parallel sides AB and CD, let E be the intersection of the diagonals AC and BD. Thus by AA, the triangles ABE and CDE are similar. By the diagonals are transversals, so the marked angles are equal: angle BAE = angle DCE and angle ABD = angle CDE. Construct point E as the intersection of the diagonals AC and BD. In the event we wish to distinguish trapezoids with exactly two parallel sides, we will call such trapezoids strict trapezoids. In Math 444 the official definition of a trapezoid is the Inclusive Definition. However, it is important to have agreement in a math class on the definition used in the class. ![]() It is possible to function perfectly well with either definition. This fits best with the nature of twentieth-century mathematics. The advantage of the inclusive definition is that any theorem proved for trapezoids is automatically a theorem about parallelograms. This seems to have been most important in earlier times. The advantage of the first definition is that it allows a verbal distinction between parallelograms and other quadrilaterals with some parallel sides. Trapezoids and under the first, they are not. The difference is that under the second definition parallelograms are However, most mathematicians would probably define the concept with theĪ quadrilateral having at least two sides parallel is called a Two and only two sides parallel is called a trapezoid. In B&B and the handout from Jacobs you got the Exclusive Definition. Believe it or not, there is no general agreement on the definition of a trapezoid.
0 Comments
Leave a Reply. |
AuthorWrite something about yourself. No need to be fancy, just an overview. ArchivesCategories |