Sample Problems for Exam 3
Math 365

Here are some sample problems. I also recommend doing the review problems at the end of each of the two chapters. Also, make sure that you have done the homework problems.

1. Reproduce
   

   

2. Construct an equilateral triangle with the following side

   

3. Construct a angle, without using a protractor.
4. Construct an isosceles triangle with as the angle between the two congruent sides.
5. Given below, construct so that the measure of is the measure of plus measure .
   

   

6. Construct the following if possible. (you are welcome to use a protractor to draw the original angles)
   1. A triangle with angles , and included side of 6 cm.
   2. A triangle with angles and and non-included side of 6 cm on the angle.
   3. A right triangle with one angle measuring and a leg of 5 cm on one leg of the acute angle.
   ex
7. Which of the previous have more than one solution.
8. Draw two quadrilaterals with two angles and the included side are congruent but the quadrilaterals are not.
9. Draw a Rhombus, connect opposite vertices.

   

   1. Argue that the line bisects and .
   2. Argue that bisects and also bisects .
10. Draw a rectangle. Connect the midpoint of each side with the adjacent midpoints.

    

    1. What figure is formed?
    2. Prove it.
11. Draw all of the altitudes to . em

    

12. Draw a triangle. Bisect each side. Extend each bisector until they intersect. Label that point O. This is the center of a circle that has interesting properties relative to the triangle. Find the property.
13. Draw a triangle. bisect each angle. Extend each bisector until they intersect. Label that point O. This is the center of a circle that also has interesting properties. What is it?
14. Using a compass and straight-edge construct the following if possible. If there is more than one possible answer draw another.
    1. A square given 1 side.
    2. A square given 1 diagonal.
    3. A rectangle given 1 diagonal.
    4. A rectangle given 2 sides.
    5. A parallelogram given 2 adjacent sides.
    6. A rhombus given the two diagonals.
    7. An isosceles triangle given the base and the angle opposite the base. (sort of hard)
    8. A trapezoid, given four sides. (hard)
15. Draw a line segment. Draw two points not on the line segment. Construct a circle that has a center on the line segment and the circle passes through both of the other two points.
16. Construct a 12-sided regular polygon.
17. Identify the congruent triangles and show they are congruent. Note that a picture can have more than one pair of triangles.

    1. 

    2. 

    3. 

    4. 

    5. 

    6. 

    7. 

    8. 

18. Draw a line segment. Draw a point not on the line segment. Construct a line parallel to the line segment that passes through the point. Now draw a line perpendicular to the line segment that passes through the point. Repeat the second part of this exercise with a point that lies on the line segment.
19. Are the following figures necessarily similar? If so explain why, if not, draw an example to show why not.
    1. Any two equilateral triangles.
    2. Any two isosceles triangles.
    3. any two right triangles having an acute angle of measure
    4. Any two isosceles right triangles.
    5. Any two congruent triangles.
    6. A triangle with sides of lengths 3 and 4 and an angle of and a triangle with sides of lengths 6 and 8 and an angle of .
20. Problems 10 and 11 in the review exercises for Chap. 13. (Chap. 11 in old book)
21. Plot the points on a Cartesian coordinate system. (Graph paper is helpful, though not absolutely necessary.)
    1. Connect these points, in the given order, with line segments.
    2. Find the midpoint of the line segment .
    3. Find the point on the line segment which is of the distance from T to U.
22. Consider the quadrilateral and D(8,0). Let P,Q,R, and S be the midpoints of respectively. Sow that the segments and bisect each other.

    Now draw the quadrilateral ABCD and the segments and on a coordinate system.

23. Graph each of the following lines on a coordinate system.
    1. 2y-16=0
    2. x=-7
    3. x+y=2
    4. 3x+5y=12
    5. 6x+10y=24
    6. y=3x+4
24. Find the equation of the line that passes through the two points (1,-2) and (4, 1). Then find the line that is perpendicular to that line which passes through the point (2,0). Find the line parallel to the first line that passes through (2,0).
25. Are the lines 3x+7y+15=0 and 6x+15y=-31 parallel? Perpendicular? Neither?
26. In words, describe the set of points satisfying each of these equations.
    1. 
    2. 
    3. 
    4. 
27. Write the equations of the circles satisfying these conditions.
    1. Center at (2,5), radius 3
    2. Center at (3,-4), radius 1
28. Show that the perpendicular bisectors of the sides of a triangle are concurrent. (Using coordinates.)
29. Problem 17 page 914. (prob 18, page 942 in old book)
30. Graph each of these functions.
    1. 3x+2y+6=0
    2. 
    3. 
    4. 
31. Populations of bacteria grow because, after a suitable interval, each cell divides into two identical cells. A certain type of cell divides once every minute.
    1. If one such cell is placed in an agar dish at time t=0, what function, p(t), gives the population size at the end of t minutes?
    2. Graph the function p(t) of part (a) using suitable scales on the vertical and horizontal axes.
32. Graph the function and determine the minimum value of the function and the value of x for which it occurs.

• About this document ...