B&B, p. 75 - #10,11,12. Suggestion: Read over all 3 problems and think about them before you begin. Can you combine arguments to cover all cases in one proof? Do you have to do more geometry for 12 or just use logic?
Note: Do not use area in your proof. The area formula you want to use will be a consequence of this problem, so you cannot assume it.Let ABC be an isosceles triangle (AB = AC). Suppose there is a point D on segment AB so that CD = CB. Let us label the lengths: AB = AC = x and CD = CB = y.
(a) With these assumptions what is length BD in terms of x and y? Also, write the length AD in terms of x and y.
(b) If we make the additional assumption that triangle DAC is also isosceles (i.e., DA=DC), find the ratio r = x/y. The answer should be a real number (no unknowns). Hint: If r = x/y, then x = ry, so you can substitute for x in your equation.
(c) Continuing with the assumption of (b), let angle BAC = a and let angle ABC = b. Find all the other angles in the figure in terms of a and b. Is it possible to derive a numerical value of a and b in this case?
Suppose OAB is a triangle with |OA| = 5, |OB| = 4, and |AB| = k, let A' be the point on OA with |OA'| = 1/5 and let B' be the point in OB with |OB'| = 1/4. Find the distance |A'B'| (this will be a formula involving explicit real numbers and k, no other quantities). Show your reasoning.
On a paper, draw a segment. Call the length of the segment d. With straightedge and compass, construct d times the square root of 5 two different ways.