Geometry Midterm Study Guide
Name _________________________________
Tips on how to do well on the midterm:
1) Look over the following outline.
2) Do the practice problems included in this study guide.
3) Review your portfolio/study sheets from each unit.
4) Look over unit tests.
Outline:
Unit 1: Coordinate Geometry
 Linear Equations – know about slopes of parallel and perpendicular lines; y=mx+b
 Quadratic/Linear Systems – graph and find solutions
 Circle/Linear Systems – graph and find solutions; know equation of a circle
 Formulas – distance, midpoint, slope
 Properties of Triangles – classification by angles and sides; algebraic problems
 Properties of Quadrilaterals – used for algebraic problems
 Analytical Proofs – know how to prove each type of triangle and quadrilateral
Unit 2: Similarity of Triangles
 Ratios of Similar Figures - sides, areas, perimeters, volumes, etc.
 Mid-segment of Triangle theorem
 Mean proportionality - geometric mean, leg theorem, altitude theorem
 Ratios of Special Right Triangles
Unit 3: Constructions
 Basic constructions
 Concurrences – centroid, orthocenter, incenter, circumcenter, locations and theorems of each
Unit 4: Locus
 5 basic theorems
 Compound Locus – on and off coordinate plane; writing equations of locus
Unit 5: Reasoning (Logic)
 Statements; Negations; Conjunctions; Disjunctions; Truth Values
 Conditionals; bi-conditionals; inverse; converse; contrapositive; logically equivalent
Unit 6: Triangle Congruence
1. Properties of Triangles – interior angle theorem, triangle inequality theorem; exterior angle theorem
2. Side-angle relationships; exterior angle inequality theorem
3. Vocabulary – altitude, median, perpendicular lines, angle and segment bisectors, midpoint, etc.
4. Triangle Congruence Theorems – Proofs using SSS, SAS, ASA, AAS
Practice Problems:
Unit 1
1. The endpoints of
are
2. M is the midpoint of
the coordinates of B?
and
. What are the coordinates of the midpoint of
. If the coordinates of A are
3. The coordinates of point R are
and the coordinates of M are
and the coordinates of point T are
4. What is the slope of a line perpendicular to the line whose equation is
5. Two lines are represented by the equations
and
?
, what are
. What is the length of
?
?
. For which value of m will the lines
be parallel?
6. What is an equation of the line that passes through the point
equation is
?
and is perpendicular to the line whose
7. On the set of axes below, solve the following system of equations graphically for all values of x and y.
8. What is an equation of a circle with its center at
and a radius of 4?
9. Solve the following system of equations:
x  22   y  32  9
x 1
10. In the accompanying diagram of parallelogram ABCD,
of degrees in
.
and
. Find the number
11. In parallelogram STUV,
,
, and
. What is the length of
12. Given:
,
,
Prove:
is an isosceles right triangle.
[The use of the grid is optional.]
13. Given:
,
,
,
Prove: ABCD is a parallelogram but not a rectangle. [The use of the grid is optional.]
?
Unit 2
14.
is similar to
,
,
,
, and
. What is the length of
?
15. The accompanying diagram shows two similar triangles. Find the value of x.
16. Delroy’s sailboat has two sails that are similar triangles. The larger sail has sides of 10 feet, 24 feet, and
26 feet. If the shortest side of the smaller sail measures 6 feet, what is the perimeter of the smaller sail?
17. The ratio of the corresponding sides of two similar squares is 1 to 3. What is the ratio of the area of the
smaller square to the area of the larger square?
18. Find the value of x.
19. In ΔABC, ∠A is a right angle and m∠B = 45. If AB = 20 ft, find BC.
20. Find the values of x and y given that
21. In the diagram below of
perimeter of
.
,
||
.
is a midsegment of
22. In the diagram below of right triangle ACB, altitude
the length of
in simplest radical form.
,
intersects
,
, and
at D. If
. Find the
and
, find
23. In the diagram below, the length of the legs
and
of right triangle ABC are 6 cm and 8 cm,
respectively. Altitude
is drawn to the hypotenuse of
. What is the length of
to the nearest
tenth of a centimeter?
Unit 3
24. Using a compass and straightedge, construct the bisector of the angle shown below. [Leave all construction
marks.]
25. Using a compass and straightedge, and
[Leave all construction marks.]
below, construct an equilateral triangle with all sides congruent to
26. Using only a compass and a straightedge, construct the perpendicular bisector of
[Leave all construction marks.]
27. In the diagram below of
.
, medians
,
, and
intersect at D, and
and label it c.
. Find the length of
.
28. Which geometric principle is used in the construction shown below?
1)
2)
3)
4)
The intersection of the angle bisectors of a triangle is the center of the inscribed circle.
The intersection of the angle bisectors of a triangle is the center of the circumscribed circle.
The intersection of the perpendicular bisectors of the sides of a triangle is the center of the inscribed circle.
The intersection of the perpendicular bisectors of the sides of a triangle is the center of the circumscribed
circle.
Unit 4
29. Towns A and B are 16 miles apart. How many points are 10 miles from town A and 12 miles from town B?
30. The distance between parallel lines and m is 12 units. Point A is on line . How many points are equidistant
from lines and m and 8 units from point A.
31. What is the total number of points equidistant from two intersecting straight roads and also 300 feet from the
traffic light at the center of the intersection?
32. How many points are equidistant from two parallel lines and also equidistant from two points on one of the lines?
33. On the set of axes below, sketch the points that are 5 units from the origin and sketch the points that are 2 units
from the line
. Label with an X all points that satisfy both conditions.
Unit 5
34. What is the negation of the statement “I am not going to eat ice cream”?
35. The statement
1) 1
2) 10
3) 5
4) 4
is true when x is equal to
36. Given: Two is an even integer or three is an even integer.
Determine the truth value of this disjunction. Justify your answer.
37. Given the statement: "If x is a rational number, then
is irrational.” Which value of x makes the
statement false?
1) 3/2
2) 2
3) 3
4) 4
38. Given the statement: “If two lines are cut by a transversal so that the corresponding angles are congruent,
then the lines are parallel.” What is true about the statement and its converse?
1) The statement and its converse are both true.
2) The statement and its converse are both
false.
3) The statement is true, but its converse is
false.
4) The statement is false, but its converse is
true.
39. Which statement is expressed as a biconditional?
1) Two angles are congruent if they have the
same measure.
2) If two angles are both right angles, then they
are congruent.
3) Two angles are congruent if and only if they
have the same measure.
4) If two angles are congruent, then they are
both right angles.
40. What is the inverse of the statement “If it is sunny, I will play baseball”?
1) If I play baseball, then it is sunny.
2) If it is not sunny, I will not play baseball.
3) If I do not play baseball, then it is not sunny.
4) I will play baseball if and only if it is sunny.
41. Which statement is logically equivalent to “If I eat, then I live”?
1) If I live, then I eat.
2) If I eat, then I do not live.
3) I live if and only if I eat.
4) If I do not live, then I do not eat.
Unit 6
42. What is the measure of the largest angle in the accompanying triangle?
43. If the measures of the angles of a triangle are represented by
1) an isosceles triangle
2) a right triangle
3) an acute triangle
4) an equiangular triangle
,
, and
, the triangle is
44. In ABC, mA  95, mB  50, and mC  35. Which expression correctly relates the lengths of
the sides of this triangle?
(1) AB < BC < CA (3) AC < BC < AB
(2) AB < AC < BC (4) BC < AC < AB
45. In
,
and
46. In the diagram below of
and
. Find
. Find
, side
.
.
is extended through P to T,
,
47. Which set of numbers represents the lengths of the sides of a triangle?
1)
2)
3)
4)
48. In the diagram of
and
below,
Which method can be used to prove
49. Given:
Prove
and
, C is the midpoint of
,
, and
?
and
.
,
50. Given: K is the midpoint of YU ; YU  KC
Prove: YKC  UKC
Y
K
C
U