Geometry of the 1950's (continued)
Many math teachers claim that geometry has changed over the last halfcentury. 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the advent of integrated math and Common Core Standards, some (including myself) feel that the meat of geometry has been replaced by a utilitarian skeleton that does little to satisfy.
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 60year period.
Part II (Part I appeared last week)
Answer three questions from Part II. Values in square brackets are credit values.
 Prove: Two right triangles are congruent if the hypotenuse and a leg of one are equal to the hypotenuse and a leg of the other. [10]
 Diagonal AC of parallelogram ABCD is extended through A to point E and through C to point F, making CF equal to AE. Lines ED, DF, FB, and BE are drawn. prove that EDFB is a parallelogram. [10]
 Prove: If two chords intersect within a circle, the product of the segments of one is equal to the product of the segments of the other. [10]
 In triangle ABC, sides AB and AC are equal. A line through B intersects AC at D. BD is extended through D to point E, and CE is drawn. prove that BE is greater than CE. [10]
 Paralell lines r and s are d distance apart and point P is any point between the two lines.
 What is the length of the radius of a circle that is tangent to both r and s? [1]
 What is the locus of the center of a circle that is tangent to both r and s? [2]
 What is the locus of the center of a circle whose radius is d/2 and which passes through P? [2]
 On your answer paper draw the figure at the right. [Construction of r parallel to s is not required.] Now construct a circle tangent to the two parallel lines and passing through the given point P. [4]
 How many different circles are there that satisfy the requirements specified in part 3? [1]
Part III
Answer two questions from Part III.
 In the accompanying figure, BCD is a semicircle and AB and ED are equal quadrants. (A quadrant is a quarter of a circle.) AE is 42 inches and the diameter of the semicircle is 28 inches. Find the area of the figure. [Use π = 22/7] [10]
 The longer base of an isosceles trapezoid is 40, one leg is 20 and one of the base angles is 63^{o}.
 Find to the nearest integer: (a) the altitude of the trapezoid [3] and (b) the shorter base of the trapezoid [4]
 Using the results found in answer to part 1, find the area of the trapezoid. [3]
 Quadrilateral ABCDE is inscribed in a circle and arcs AB, BC, CD, and DA are in the ratio 3:3:4:2.
 Find the number of degrees in each of the fouyr arcs. [2]
 If AC is drawn, find the number of degrees in angles CAB and ACD. [2]
 If chord AD is 10, find the (a) circumference of the circle, and (b) area of triangle ABC. [3]
 For each statement listed, the numerical value, correct to the enarest tenth, is given in the list below (a)(g). List the numbers 15 on your answer paper and after each number write the letter indicating the numerical value of the corresponding statement. [10]
 The side of a square whose diagonal is 10
 The area of a rhombus whose diagonals are 4.8 and 4.
 The hypotenuse of a right triangle whose legs are are 6 and 14.
 The area of a regular hexagon whose side is 2.
 The length of an arc of 80^{o} in a circle whose circumference is 37 1/2.
(a) 7.1 (b) 8.3 (c) 9.6 (d) 10.4 (e) 13.4 (f) 15.2 (g) 15.3
Note: Something odd is that this ARCO exam supposedly is from January 1954, yet the copyright of the text providing the exam has a copyright of 1950!
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!
