Geometry of the 1950's
Many math teachers claim that geometry has changed over the last half-century. In fact, given the advent of integrated math and Common Core Standards, some (including myself) feel that the meat of geometry has been replaced by a utilitarian skeleton that does little to satisfy.Part I (Parts II and III will appear next week)
Answer all questions in this part. Each correct answer will receive 2 credits. No partial credit will be allowed.
As a a means to reflect on whether or not geometry has changed, consider the following AMSCO Exam from January 1954. Can you (or your students) successfully pass it?
Note: I am not saying that this exam represents the geometry I believe should be taught (As it doesn't)...but it does provide a contrast from the geometries taught separated by a 60-year period.
Directions: Write on the line at the right of each question the expression that, when inserted in the blank, will make the statement true.
- In parallelogram ABCD, angle A is twice angle B. Find the number of degrees in angle B.
- Find the sum of the interior angles of a polygon of 12 sides.
- Is there a regular polygon such that each exterior angle is 50o? [Answer yes or no.
- The hypotenuse of a right triangle is 2.5 and one leg is 1.5. Find the other leg.
- If a point is equidistant from the three sides of a triangle, it must be the intersection of the three (a) medians, (b) angle bisectors, (c) perpendicular bisectors of the sides. which is correct: a, b, or c?
- In triangle ABC a line parallel to AC intersects AB at D and CB at E. If AB = 12, DB = 8, and AC = 15, find DE.
- The areas of two similar triangles are 40 and 90. Find the ratio of a side of the smaller polygon to the corresponding side of the larger.
- A tangent and a secant are drawn to a circle from the same point are 6 and 18 respectively. Find the external segment of the secant.
- The segmenst made by the altitude on the hypotenuse of a right triangle are 3 and 7. Find the altitude on the hypotenuse. [Answer may be left in radical form.]
- An angle of a rhombus is 60o and its shorter diagonal is 4. Find the altitude of the rhombus. [Answer may be left in radical form.]
- The locus of points equidistant from two concentric circles whose radii are 8 and 14 is a circle concentric with the given circles. Find its radius.
- Two unequal circles are tangent externally to the same line at the same point. From any point in this line one tangent is drawn to the larger circle and one tangent is drawn to the smaller circle. The tangent to the larger circle is (a) longer than, (b) equal to, (c) shorter than, the tangent to the smaller circle. Which is correct: a, b, or c?
- Express the area of a regular polygon in terms of the number of sides (n), the length of one side (s), and the apothem (a).
- In triangle ABC, angle C = 90o, AC = 20 and BC = 36. Find angle A to the nearest degree.
Directions: For each of the following, if the statement is always true, write the word true on the line to the right; if it is not always true, write the word false.
- If a line a is perpendicular to line b and line c is parallel to a, then c is ____ to b.
- An angle formed by two chords intersecting within the circle is measured by one half the ____ of the intersected arcs.
- If the cnter of the circle that is circumscribed about a triangle is outside the triangle, the triangle is _____.
- If two parallelograms have equal bases, their areas are to each other as their ____.
- It is alwasy possible to construct a right triangle if the given parts are the hypotenuse and one of the acute angles.
- An equilateral polygon inscribed in a circle is regular.
- If the diagonals of a parallelogram are equal, the parallelogram is a square.
- Ab and A'B' are bases of isosceles triangles ABC and A'B'C'. If AB:A'B' = AC:A'C', the triangles are similar.
- If the sides of a triangle are a, b, and c, the perimeter of the triangle formed by joining the mid-points of the sides of the given triangle is (a+b+c)/2.
- The median drawn to to the hypotenuse of a right triangle divides the triangle into two isosceles triangles.
- The accompanying diagram shows the division of given line segment AB into two parts which are in the ratio r:s. Which statement, 1 or 2, is used to prove that the cosntruction is correct?
Note: Parts II and III of this ARCO exam (1954) will be appear next week.
- If a line is drawn through two sides of a triangle parallel to the third side, it divides the sides proportionally.
- A line that divides two sides of a triangle proportionally is parallel to the third side.
Source: I. Dressler, Reviewing Plane Geometry. AMSCO, 1950
Hint: None provided, except think like a student in 1954!
Solution Commentary: None provided....for each question, you can search out a reasonable answer...even now in the year 2014!