Congruent Figures 4-7

Two figures are congruent if they have the same size and same shape. If we could lift on of the figures and place it directly on top of the other... all three vertices would match up with the others. In the example in our book ( page 132) we have two congruent triangles ∆ABC and ∆XYZ A would fall on X, B would fall on Y and C would fall on Z. These matching vertices are called corresponding vertices. Angles at corresponding vertices are corresponding angles and the sides joining corresponding vertices are corresponding sides.
The book states Corresponding angles of congruent figures are congruent.
and
Corresponding sides of congruent figures are congruent.
In class we discussed how to abbreviate the above -- when dealing with triangles.
CPCTC
Corresponding PARTS of congruent triangles are congruent!!

When we name two congruent figures we list corresponding vertices in the same order.

∆ABC ≅ ∆XYZ or ∆CAB ≅ ∆ZXY or ∆BCA ≅ ∆YZX

we know that
∠A ≅ ∠X and ∠B ≅ ∠Y and ∠C ≅ ∠Z
and the segments ( which are denoted with a line (but w/o arrows)above each of the two letters
AB ≅ XY and BC ≅ YZ and CA ≅ ZX

If two figures are congruent, we can make the coincide -- occupy the same place-- by using one or more of the following basic rigid motions:

Translation or slide
Rotation
Reflection or flip or mirror
Check the book on page 133 for good examples of these three rigid motions... I like to think of Tetris moves!!

Math 6H ( Periods 3, 6, & 7)

Circle 4-5

A circle is the set of all points in a plane at a given distance from a given point O ( called the center)
A segment joining the center to a point on the circle is called a radius (plural: radii) of the circle ALL radii of a given circle have the same length and this length is called the radius of the circle
A segment joining two points on the circle is called a chord. However a chord passing through the center is called a diameter of the circle. The ends of the diameter divide the circle into two semicircles. The length of the diameter is called the diameter of the circle.

The perimeter of a circle is called the circumference

The quotient
circumference ÷ diameter
is the same for all circles regardless of their size. This quotient is denoted by the Greek letter ∏ ( pronounced 'pie') No decimal gives ∏ exactly.-- but a good approximation is 3.14

If we denote the circumference by C and the diameter by d
we have
C ÷ d = ∏.
We can manipulate this formula to several useful forms:

C = ∏d
or d = C ÷ ∏ .
In addition since we know that d = 2r we can also have the following:

C = 2 ∏ r
or r = C ÷(2∏)

When using the approximate 3.14 for ∏ , give your answer in three digits-- as the book explains.
In addition, use the approximately equal to symbol ≈ when you have replaced ∏ .

A polygon is inscribed in a circle if all of its vertices are on the circle. See the diagram on Page 129 for a good example of an inscribed triangle.

It can be shown that three points NOT ON A LINE determine a circle.

Math 6H ( Periods 3, 6, & 7)

Polygons 4-5

A polygon is a closed figure formed by joining segments—the sides of the polygons at that endpoints—the vertices of the polygon. Polygons are names according to the number of sides they have.

Triangle 3 sides
Quadrilateral 4 sides
Pentagon 5 sides
Hexagon 6 sides
Octagon 8 sides
Decagon 10 sides

A polygon is REGULAR if all its sides and all its angles are congruent.
A regular triangle is the equilateral triangle
A regular quadrilateral is the square.

To name a polygon we name its consecutive vertices in order.
A diagonal of a polygon is a segment joining two non consecutive vertices.

To find the perimeter of a polygon add all the lengths of its sides. The perimeter is the distance around the figure. Finding the perimeter of a parallelogram can be done by computing the sum of the lengths or by using the distributive property to obtain
For instance a parallelogram with sides 9 cm and 6 cm
has a perimeter of 9 + 6 + 9 + 6 = 30 cm
but you could calculate that by 2(9) + 2(6) = 18 + 12 = 30 cm or using the distributive property, even 2(9+6) = 2(15) = 30 cm

If you have a regular polygon you can simple multiple the side by the number of sides in the polygon
For example,
a quadrilateral with side 16.5 m has a perimeter of 4(16.5) = 66 m

The sum of the measures of the angles of any pentagon is 540 degrees. If it is a regular pentagon, what must be the measure of each angle of the regular pentagon? 540/5 = 108 degrees.

The sum of the measures of the angles of any pentagon is 540 degrees. How can you prove that? Draw your pentagon and then draw all the diagonals from ONE of the vertices. Count the number of triangles created. Three. How many degrees does a triangle have? 180. Multiply the number of triangles created by 180… 540 is your answer. IT works every time. So How could you create a general rule or formula for the sum of the measures of the angles of any polygon with n sides?

Practice drawing various polygons—now practice drawing all the diagonals for each of them. Can you determine a general rule for the number of diagonals that can be drawn for any polygon?