Study Sheet for Quiz 3

Reciprocal lengths and Similar Triangles

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.

Constructing the Square Root of 5

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.

• First, construct a right triangle with legs a and b and hypotenuse d*sqrt 5.
• Second, construct a segment AB of length 6d and mark C as the point on AB with AC = d. Then CB = 5d. Carry out the construction of the mean proportional of AC and CB. Important: This should be a method different from the first method. It should work for any numbers such as d and 7d, or two segments a and b drawn on the page, as well as for d and 5d. You will not be able to use the simple right triangle method above for 7. Reference: Page 67 of Berele-Goldman.

Constructing Cross-Sections and Dihedral Angles

Be able to construct plane cross-sections of these shapes that could be used to measure dihedral angles.       

• Regular Tetrahedron
• Pyramid with Square Base and Equilateral Triangle Sides

Isometries

• Be able to state the definitions and state the theorems from the assigned reading in Brown.
• For lines m and n and a point P, be able to construct R_mR_n(P).  (Note: Order is important.)