Midterm Review Key
1. The slopes of perpendicular lines are negative reciprocals. So you need to find the slope of the
given line and then flip and negate it. You are given:
5x + 3y = 8 so solve for y and pick out the slope by
-5x
-5x
3y = -5x + 8
3
3
3
Y=
−5
x
3
8
+3
which has a slope =
−5
,
3
the negative reciprocal is
Therefore, the slope of the perpendicular line =
+3
5
3
5
2. Parallel lines have the same slope. Given the equation y= 2x +1. A line that is parallel to it would
have the same slope so you need to use a slope = 2 and through the point (2, -1).
When given a point and a slope you can solve for the y-intercept by using the point as x and y
and using the slope as m in y = mx +b.
So, if y = mx + b
-1 = 2(2) + b
-1 = 4 + b
-4 -4
-5 = b
Therefore, the standard form of the equation of the line parallel to y = 2x + 1 and through the
point (2, -1) is y = 2x – 5.
3. The midpoint = (
𝑥1+ 𝑥2 𝑦1+ 𝑦2
,
2
2
) so given endpoints A(7, -1) and B(-3, 3) substitute into the
midpoint formula: midpoint = (
7+ −3 −1+3
,
)
2
2
4 2
2 2
= ( , ) = (2, 1).
4. If A(-3, 6) and the midpoint = (-5, 2), you can solve for the other endpoint by setting up separate
equations to solve for the x and y parts of the coordinate for point B as follows:
𝑥1+ 𝑥2
So
Solve
So
2(-5)
-3 + x = -10
+3
𝑦1+ 𝑦2
= x part of the midpoint
2
−3+𝑥
= -5
2
−3+𝑥
for x: 2( 2 ) =
+3
X = -7
Solve for y:
= y part of the midpoint
2
6+𝑦
=2
2
6+𝑦
2( 2 ) =
2(2)
6+y=4
-6
-6
y = -2
Therefore, the midpoint = (-7, -2). So, the answer for number 4 is choice 4.
5. Distance = √(𝑥1 − 𝑥2 )2 + (𝑦1 − 𝑦2 )2 = √(−1 − 3)2 + (4 − −2)2 = √(−4)2 + (6)2 =
√16 + 36 = √52 = √4 x √13 = 2√13. Final answer 2√13.
See the construction review packet for reference on any construction question
6. Choice 3 is the full illustration of the construction of an angle bisector. The other choices show
incomplete illustrations of the construction.
7. Choice 1 is the complete illustration of the construction of the perpendicular bisector.
8. The illustration shows congruent corresponding angles. Choice 4 says “When two lines are
intersected by a transversal and the corresponding angles are equal, the lines are parallel.”.
Therefore, Choice 4 is the answer (the other choices don’t even mention congruent
corresponding angles).
9. Choice 1 is the correct illustration.
10. The sum of the angles of a triangle = 180. So add the three expressions that are given and set
the sum = 180. (3x + 1) + (4x -17) + (5x – 20) = 180
3x + 4x + 5x + 1 – 17 – 20 = 180
12x – 36 = 180
+ 36 +36
12x = 216
12
12
X = 18
3x + 1 = 3(18) + 1 = 55
4x – 17 = 4(18) -17 = 55
5x – 20 = 5(18) -20 = 70
Since two angles = 55, the triangle is isosceles and choice 3 is the correct answer.
(Note: a right triangle has a 90 degree angle, a scalene triangle has 3 different degree measures,
and an equilateral triangle has all 3 angles = 60)
11. According to the diagram x = the vertex angle and 42 and y are the congruent base angles of the
given isosceles triangle. So y = 42. If 42 + 42 + x = 180
84 + x = 180
-84
-84
Then, X = 96
So x = 96 and y = 42. Choice 4 is the correct answer.
12. The sum of the remote interior angles = The exterior angle of the triangle
So, <A + <B = <ACD
(x) + (2x +15) = 5x + 5
3x + 15 = 5x + 5
-3x
-3x
15 = 2x + 5
-5
-5
10 = 2x
2 2
5=x
So, <B = 2x + 15 = 2(5) + 15 = 25.
Choice 3 is the correct answer.
13. The sum of the two smaller sides of the triangle must be greater than the length of the third
side. So check the choices: 1) 2 + 4 is not greater than7 so a triangle cannot be made
2) 4 + 5 is greater than 6 and all three sides are different so this is the
correct choice. It is possible and it is scalene.
Choice 2
14. The sides opposite the smallest and largest angles correspond to the smallest and largest sides
of the triangle respectively. We first need to find the measure of all three angles of the triangle,
which must total 180 degrees. So, 70 + 65 + x = 180
135 + x = 180
-135
-135
X = 45
B
65
smallest side
largest side
A
45
70 C
Medium Side
So, BC < AC < AB is the correct order small to large
Choice 3 is correct
15. Remember “My Parents Are Aliens”
The circumcenter is found by constructing the perpendicular bisectors. So , choice 4 is correct.
16. The centroid divides the parts of each median in a 2:1 ratio.
BF = 18 and is the length of the entire median so if
So, choice 4 is correct.
18
3
= 6, then PF = 6 and BP = 12.
17. The sum of the interior angles of a polygon = 180(n – 2)
(n represents the number of sides)
A hexagon has 6 sides so, 180(6 -2) = 720 total
We have 5 of the 6 angles 150 + 100 + 80 + 165 + 150 = 645
720 – 645 = 75 degrees for the remaining angle. Choice 1 is correct.
18. Each exterior angle of a regular polygon =
360
𝑛
=
360
5
= 72
So, choice 2 is correct.
19. Each interior angle of a regular polygon =
In this case,
180(𝑛−2)
𝑛
180(𝑛−2)
𝑛
= 120 and we must solve for n.
First cross multiply so that,
Then distribute the 180
20.
21.
22.
23.
24.
25.
180(n – 2) = 120n
180n – 360 = 120n
-120n
-120n
60n – 360 = 0
+ 360 +360
60n = 360
60
60
n = 6 sides
If you plot (5, 2) on graph paper and rotate the paper 90 degrees in a counterclockwise
direction, you will notice that the image of the point (5, 2) is now at (-2, 5). So, choice 3 is
correct. (Alternatively, you can use the rule (x, y) → (-y, x) for a 90 degree counterclockwise
rotation).
If you plot the point (-3, 7) and reflect it across the x-axis you will see that the image of the point
will now be at (-3, -7). Choice 2 is correct. (Alternatively, you can use the rule (x, y) → (x, -y)
when reflecting the x-axis).
If you plot (3, 4) and reflect it across the y-axis you will see the image of the point will now be at
(-3, 4). Choice 2 is correct. (Alternatively, you can use the rule (x, y) → (-x, y) when reflecting the
y-axis).
Remember, it is difficult to count boxes diagonally across the line y = x so we used the rule:
(x, y) → (y, x) when reflecting across the y = x line. In this case, (3, -4) → (-4, 3). Choice 4 is
correct.
If you plot the point (-3, -1) and rotate the paper 180 degrees you will see that the image of the
point will now be at (3, 1). Choice 1 is correct. (Alternatively, you can use the rule (x, y) → (-x, -y)
for a rotation of 180 degrees).
Translations are achieved by adding the corresponding x and y parts of the translation vector to
the x and y parts of the pre-image coordinate. In this case we must first figure out what
translation vector would move P(3, 5) to 𝑃′ (6, 1).
To get from 3 to 6 we must add 3.
To get from 5 to 1 we must subtract 4.
So, the translation vector used is <3, -4>.
Now translate (-3, -5) along the vector <3, -4>. -3 + 3 = 0 and -5 + -4 = -9
So the image of the point winds up at (0, -9). Choice 1 is correct.
26. Starting at (4, 2) and first reflecting y=x (like #23) brings you to (2, 4) and then rotating that
point by 90 degrees in the counterclockwise direction (like #20) brings us to a final image point
of (-4, 2). Choice 1 is correct.
27. A dilation does not preserve distance so that is incorrect.
A translation preserves both length and orientation so this is the correct answer. Choice 2.
(Note: Line reflections and glide reflections do not preserve orientation.
Second 27. Dilations change the size of the figure. So Choice 3 is correct.
28. Right 3 is achieved by adding 3 to x and down 7 is achieved by subtracting 7 from y.
So, (x + 3, y – 7). Choice 1 is correct.
(Note: as a vector (x + 3, y – 7) may be written as <3, -7>)
29. A median connects a vertex of a triangle to the midpoint on the opposite side. A midpoint
divides a line segment into two congruent parts. The only choice that illustrates this point is
choice 1.
30. A perpendicular bisector forms a 90 degree angle at the midpoint of a line segment.
So, AC ≅ DC because AD is bisected at point C.
BC may or may not be congruent to CD . Choice 2 is correct then.
(Note: the angles in choice 3 are both 90 so they are congruent and the triangles can be proven
congruent by SAS)
31. The negation of “Squares are parallelograms.” Is “It is not the case that squares are
parallelograms.”. Choice 3 is correct.
Second 31.
An “AND” (Conjunction) statement is only true if both parts are true so we need a number that is not
the square of an integer and a multiple of 3. 18 is a multiple of 3 and it is not the square of an integer so
this is the correct answer. Choice 2. (Note: 9 IS the square of an integer, 32 is NOT a multiple of 3, and
36 IS the square of an integer)
32. An “OR” (Disjunction) statement is false only if both parts are false. We need a number that
cannot be divided by 2 or 3. 11 is the only number that cannot be divided by 2 or 3 so this is the
correct answer. Choice 3. (Note: 6 is divisible by both 2 and 3, 8 is divisible by 2, and 15 is
divisible by 3)
33. If → Then (Conditional) statements are false only if the first part is true and the second part is
false. So, we need a number that is divisible by 8 but not by 6. 32 fits this scenario. Choice 3 is
correct.
34. The “Inverse” negates both parts of the conditional statement. So choice 3 is correct.
35. The “Converse” switches the order that the conditional statement is written. So choice 3 is
correct.
36. The “Contrapositive” does both switch and negate. So choice 4 is correct.
37. The “Contrapositive” is the only one that is always “logically equivalent” to the original. So,
choice 4 is correct.
38. The triangles given are already labelled so we can see that we need AG ≅ OL to give us SAS.
So, choice 2 is correct. MARK THE TRIANGLES.
39.