EDUCATIONAL GURU MATH: August 2008

Venn Diagram

4. A guidance counselor is planning schedules for 30 students. Sixteen students say they want to take French, 16 want to take Spanish, and 11 want to take Latin. Five say they want to take both French and Latin, and of these, 3 wanted to take Spanish as well. Five want only Latin, and 8 want only Spanish. How many students want French only?

Start with the Latin circle. You have sufficient information to determine how many students want both Latin and Spanish (1).
Using this new information, you can work within the Spanish circle to determine how many students want both Spanish and French (4).
Using this new information, you can work within the French circle to determine how many students want French only (7).
The last step could be represented algebraically as

x + 4 + 3 + 2 + 5 + 8 + 1 = 30
x + 23 = 30
x = 7

Venn Diagram

3. A veterinarian surveys 26 of his patrons. He discovers that 14 have dogs, 10 have cats, and 5 have fish. Four have dogs and cats, 3 have dogs and fish, and one has a cat and fish. If no one has all three kinds of pets, how many patrons have none of these pets?

7 + 4 + 0 + 3 + 1 + 5 + 1 + x = 26

21 + x = 26

x = 5 patrons have none of these animals

Must be 5 iguana lovers!

Venn Diagram Q

2. In a school of 320 students, 85 students are in the band, 200 students are on sports teams, and 60 students participate in both activities. How many students are involved in either band or sports?

25 + 60 + 140 = 225

There are 225 students involved
in either band or sports.

CAUTION!
The fact of "320 total students" mentioned in this problem was not used in the actual solution. Be sure to organize your thinking clearly. A problem may include a piece of information that is not needed to solve the problem. Beware!

Working with Venn Diagrams

In a class of 50 students, 18 take Chorus, 26 take Band, and 2 take both Chorus and Band. How many students in the class are not enrolled in either Chorus or Band?

16 + 2 + 24 + x = 50

42 + x = 50

x = 8 students

ANGLES

What is an Angle?

Two rays that share the same endpoint form an angle. The point where the rays intersect is called the vertex of the angle. The two rays are called the sides of the angle.

Example: Here are some examples of angles.

We can specify an angle by using a point on each ray and the vertex. The angle below may be specified as angle ABC or as angle CBA; you may also see this written as ABC or as CBA. Note how the vertex point is always given in the middle.

Example: Many different names exist for the same angle. For the angle below, PBC, PBW, CBP, and WBA are all names for the same angle.

Degrees: Measuring Angles

We measure the size of an angle using degrees.

Example: Here are some examples of angles and their degree measurements.

Acute Angles

An acute angle is an angle measuring between 0 and 90 degrees.

Example:

The following angles are all acute angles.

Obtuse Angles

An obtuse angle is an angle measuring between 90 and 180 degrees.

Example:

The following angles are all obtuse.

Right Angles

A right angle is an angle measuring 90 degrees. Two lines or line segments that meet at a right angle are said to be perpendicular. Note that any two right angles are supplementary angles (a right angle is its own angle supplement).

Example:

The following angles are both right angles.

Complementary Angles

Two angles are called complementary angles if the sum of their degree measurements equals 90 degrees. One of the complementary angles is said to be the complement of the other.

Example:

These two angles are complementary.

Note that these two angles can be "pasted" together to form a right angle!

Supplementary Angles

Two angles are called supplementary angles if the sum of their degree measurements equals 180 degrees. One of the supplementary angles is said to be the supplement of the other.

Example:

These two angles are supplementary.

Note that these two angles can be "pasted" together to form a straight line!

Vertical Angles

For any two lines that meet, such as in the diagram below, angle AEB and angle DEC are called vertical angles. Vertical angles have the same degree measurement. Angle BEC and angle AED are also vertical angles.

Alternate Interior Angles

For any pair of parallel lines 1 and 2, that are both intersected by a third line, such as line 3 in the diagram below, angle A and angle D are called alternate interior angles. Alternate interior angles have the same degree measurement. Angle B and angle C are also alternate interior angles.

Alternate Exterior Angles

For any pair of parallel lines 1 and 2, that are both intersected by a third line, such as line 3 in the diagram below, angle A and angle D are called alternate exterior angles. Alternate exterior angles have the same degree measurement. Angle B and angle C are also alternate exterior angles.

Corresponding Angles

For any pair of parallel lines 1 and 2, that are both intersected by a third line, such as line 3 in the diagram below, angle A and angle C are called corresponding angles. Corresponding angles have the same degree measurement. Angle B and angle D are also corresponding angles.

Angle Bisector

An angle bisector is a ray that divides an angle into two equal angles.

Example:

The blue ray on the right is the angle bisector of the angle on the left.

The red ray on the right is the angle bisector of the angle on the left.

Perpendicular Lines

Two lines that meet at a right angle are perpendicular.

Circle basics

You must be able to recognise when an equation represents a circle.

Any equation of the form

will represent a circle, provided at least one of p, q and r is not zero.

The general equation of a circle normally appears in the form

where (-g, -f) is the centre of the circle and $\sqrt {g^2 + f^2 - c}$ is the radius.

Notice that for the circle to exist

.

Look at the following worked examples.

For

so equation represents a circle with centre = (-3, 4) and radius $= \sqrt {36} = 6$

For

does not represent a circle.

For

we must write this starting

like this:

$\eqalign{ x^2 + y^2 - 2x + {1 \over 3}y - 3 = 0\cr\cr g^2 + f^2 - c = ( - 1)^2 + ({1 \over 6})^2 - ( - 3) = 4{1 \over {36}} = {{145} \over {36}}\cr$

so equation represents a circle with centre = $(1, - {1 \over 6})$ and radius $\sqrt {{{145} \over {36}}} = {{\sqrt {145} } \over 6}$

Circle basics

Circle basics: sample questions

For each of the following equations, state whether it could represent a circle and if so, state the radius and centre.

Question 1

The Solution

Step 1: $x^2 + y^2 + 2x - {3 \over 2}y - 3 = 0$

Step 2: $g^2 + f^2 - c = 1^2 + ( - {3 \over 4})^2 - ( - 3) = {{73} \over {16}}$

The Answer

so equation represents a circle with centre ${\rm{( - 1, }}{3 \over 4})$ , radius ${{\sqrt {73} } \over 4}$

Question 2

The Solution

The Answer

so equation does not represent a circle

When you are trying to build up the equation of the circle, and you know the radius and the centre, it's easier to use the equation

, where (a,b) represents the centre of the circle, and r is the radius. This equation is the same as the general equation of a circle, it's just written in a different form.

Follow the worked examples to see how this works.
Write down the equation of the circle with centre (2, -3) and radius $\sqrt 7$ .
$\left( {x - 2} \right)^2 + \left( {y - ( - 3)} \right)^2 = \left( {\sqrt 7 } \right)^2$

If required for further work you can expand this to give