Download Math 20 Module 5 Review - Westwind Alternate School

Math 20 Module 5 review
Definitions:
1) Inductive Reasoning: This is a method of logical reasoning that starts with
several examples and ends with a conclusion. Note: Because we cannot test out
every possible example the conclusion may not be true.
-Example A: 20 students in a class drew one triangle each and then used a
protractor to measure the three interior angles. The sum of these interior angles
came out to be 180 degrees for all of the students.
*Conclusion: The sum of the interior angles of every triangle is 180 degrees. This
conclusion turns out to be true.
-Example B: 3 squared is 9 , 5 squared is 25 , 7 squared is 49 , 9 squared is 81.
Conclusion: The squares of a number is always an odd number. This conclusion is
false because 4 squared is 16.
2) Deductive Reasoning: This is a method of logical reasoning that starts with
several statements known to be true and ends with a conclusion. Note: Because the
statements were known to be true the conclusion must also be true.
-Example A: Jane was born in Calgary. All Calgarians are Canadians. Conclusion: Jane
is a Canadian.
PRACTICE PROBLEMS
A. In order to show that the sum of two even integers is an even number, two students
make the following conclusions:
Student #1
2 + 4 = 6, 4 + 6 = 10, 102 + 24 = 126, 422 + 8 = 430, therefore it appears that two even
numbers have a sum that is also an even number.
Student #2
Let two even numbers be 2x and 2y. Therefore, their sum is 2x + 2y = 2(x + y), which is
a multiple of 2 and also an even number. Therefore, two even numbers always sum to be
an even number.
Which of the following is true.
A. Student #1 used inductive reasoning, Student #2 used deductive reasoning.
B. Student #1 used deductive reasoning, Student #2 used inductive reasoning.
C. Both students used inductive reasoning.
D. Both students used deductive reasoning.
3) Counter example: any example that can be used to show that a conclusion is
false.
4) Nagation: the statement that has the opposite meaning of another statement.
Example: John is not in grade 11. This is the opposite of the statement; John is in
grade 11.
5) Converse: the "reverse" of a conditional statement. Example: If x = 4 , then x + 3
= 7. This is the converse of the statement If x+3 =7 then x=4
6) Countrapositive: the "negative/reverse" of a conditional statement. Example:
If x≠4, then x≠ 3 + 7
This is the contrapositive of the conditional statement: If x = 4 , then x + 3 = 7
*Remember the difference between and & or:
and= that both statements have to be true
or= either statement could be true but not both
PRACTICE PROBLEMS
1. Match the graph to the following statements:
a) x<4 and x>-5
b) x >4 or x <-5
c) x ≥5 or x ≤-5
d) x >5 or x <-5
Things that are known to be true:
a) Sum of angles in a triangle is 180o
b) Angle at a straight line is 180 o
Example: If (b) is true then 1=140 o
c) Alternate angles (formed from a transversal of two parallel lines) are equal.
Opposite angles (formed from intersecting lines) are equal.
d) By observing two triangles, if SSS, SAS or ASA are true (remember, SAS the
angle must be the contained angle between the sides, and in ASA, the side must
be the contained side between the two angles) then the triangles are congruent.
SSS= ALL THREE SIDES
SAS= SIDE ANGLE SIDE
ASA= ANGLE SIDE ANGLE
E) Perpendicular Bisector Theorem:
Any point on the perpendicular bisector of a line segment is equidistant from the
endpoints of the segment.
f) Chord Property
A line through the center of a circle that bisects (cuts the chord into two equal segments)
a chord is perpendicular to the chord.
g) A tangent to a circle is perpendicular to the radius at the point of tangency.
This is a very useful property when the radius that connects to the point of tangency is
part of a right angle, because the trigonometry and the Pythagorean Theorem apply to
right triangles.
Triangle Types:
Acute
Right
Obtuse
Scalene
Isosceles
Equilateral
Characteristics
o
all angles are less than 90
one angle is 90o
one angle is greater than 90o
no two sides are congruent
two sides are congruent. Note: The sides opposite the equal angles are
also equal.
all three sides are congruent. Note: The sides opposite the equal angles
are also equal.
Circles
1.
A chord is a segment both of whose endpoints are points on the circle.
2.
3. An angle inscribed in a semi-circle is a right angle.
4.
The measures of a (an) inscribed angle is half the measure of a central angle
on the
same arc.
4. Inscribed angles subtended by the same arc are equal in measure.
5.
The longest chord in a circle is called a diameter.
6.
The perpendicular bisector of a chord passes through the center of a circle.
7.
Opposite angles of an inscribed/cyclic quadrilateral are supplementary.
- A quadrilateral is said to be a cyclic quadrilateral if there is a circle passing
through all its four vertices.
9.
A tangent is a line that intersects the circle in one and only one point.
10. A Radius and a tangent form a 90o angle
11. 2 tangent lines that intersect at the same point outside a circle are equal in
length.
12.The Exterior angle of a cyclic quadrilateral is equal to the inscribed angle on the
opposite side of the cord.
13. The angle between a tangent and a chord meeting the tangent at the point of
contact is equal to the inscribed angle on opposite side of the chord.
14. Regular Polygon: Polygon where all of the sides are the same length and all of
the angles are the same measure.
- Remember the sum of the interior angle of any polygon with n sides is 180o (n-2)
-The messure of each interior angle of a regular polygon with n sides is
=180o (n-2)/n
Practice Problem
Each exterior angle of a regular polygon is 15°.
a. Determine the number of sides of the polygon.
b. What is the sum of the interior angles?
Venn Diagrams
1) Kroner asked 100 adults whether they had studied French, Spanish or Japanese in
school. According to the Venn diagram below, how many had studied
a. Spanish?
b. Spanish but not French?
c. Japanese but not French?
d. French and Spanish?
e. French or Spanish?
f. French and Spanish but not Japanese?
g. How many adults speak English?
Geometric proofs
-Review the standard way to write geometric proofs
PRACTICE PROBLEMS
1. The reason for statement 4 is:
a. assumed
b. corresponding angles in congruent triangles
c. SSS congruency theorem
d. SAS congruency theorem
2. The reason for statement 5 is:
a. assumed
b. corresponding angles in congruent triangles
c. SSS congruency theorem
d. SAS congruency theorem
3. The reason for statement 6 is:
a. assumed
b. corresponding angles in congruent triangles
c. SSS congruency theorem
d. SAS congruency theorem
Circle practice problems:
Modula 5B Review
Review the following formulas:
Practice Problems
1. Write the equation of each circle with the following given information.
a. Centre is (2m,m) , radius is m.
b. Circle passes through (1,2) with it’s centre at (-3,4).
c. Both (-5,1) and (1,7) are endpoints of a diameter.
2.
3.
4. What is the equation of the tangents to the circle (x+2)2+(y+5)2=9
5. A line segment, with endpoints A( 2, 6) and B(4, 10) , is divided into 5 congruent parts.
Moving from A to B, the four division points are consecutively labelled P1 , P2 ,P3 and P4.
The coordinates of P2 are
A. ( 2.8, 7.6) B. ( 2.4, 6.8) C. ( 3.2, 7.6) D. ( 3.2, 6.8)