Download Unit 9_Basic Areas and Pythagorean theorem

UNIT 9
GEOMETRY
1.- Triangle:
A triangle is a figure formed when three non collinear points
are connected by segments.
Each pair of segments forms an angle of the triangle. The
vertex of each angle is a vertex of the triangle.
The sum of the measures of the angles of a triangle
is 180.
The sum of the lengths of any two sides of a triangle must be greater than the third side. That is:
a+b>c
a+c>b
b+c>a
The subtraction of the lengths of any two sides of a triangle must be smaller than the third side.
Triangles can be classified by:
Their sides:
Equilateral
Isosceles
Scalene
All three sides have equal
lengths
Exactly two equal sides
All sides have different
lengths
Their angles:
Acute
Right
Obtuse
All interior angles are acute
(<90º)
One angle is a right angle
(90º)
One angle is obtuse
(>90º)
Area of a triangle
There are several ways to compute the area of a triangle:
1. When you know the lenght of the base and the height, you can use the formula:
2.- Square:
A square is a parallelogram with 4 congruent sides and 4 right angles.
A square is a particular case of a rectangle and a rhombus simultaneously. So,
it shows both the properties of rhombus and rectangle simultaneously.
Square can be differentiated from a rectangle and rhombus due to following properties:
1. Unlike rectangle square needs to have all its sides equal.
2. Unlike rhombus square needs to have all angles equal to 90 degree.
Area of a square
If l is the side-length of a square, the area of the square is:
3.- Rectangle:
A rectangle is a four-sided polygon with four right angles, whose
opposite sides are parallel and are equal.
A rectangle is a particular sort of parallelogram, but can't say that all
parallelograms would be rectangles, because a rectangle is a shape
where opposites sides are parallel and all the corners are 90 degree
angles. Some parallelograms would be rectangles, but not all.
Area of a rectangle
The area of a rectangle is the product of its width and length.
4.- Rhombus:
A Rhombus is a four-sided polygon having all four sides of equal length and whose
opposite sides are parallel.
Each rhombus has two diagonals. The biggest one is called long diagonal and the
smaller one is called short diagonal. (Note that the diagonals of a rhombus are
perpendicular).
Area of a rhombus
The area of any rhombus is equal to one-half the product of the lengths D and d of its diagonals.
5.- Trapezoid:
A trapezoid is a quadrilateral with exactly one pair of parallel sides.
The parallel sides are called bases. The nonparallel sides are called
legs.
Area of a trapezoid
The area of a trapezoid is given by the formula:
, where B, b are the lengths of the two bases and h is the altitude of the
trapezoid.
6.- Parallelogram:
A parallelogram is a quadrilateral with two pairs of parallel sides.
The sum of the measures of the angles of a parallelogram is 360.
Squares, rectangles and rhombus are particular types of parallelograms.
Area of a parallelogram
The area of a parallelogram is given by the formula:
7.- Regular polygon:
A regular polygon is a polygon which is equiangular (all angles are
congruent) and equilateral (all sides have the same length).
Attributes:
•
•
•
•
•
•
The sides are the straight line segments that make up the polygon.
The vertex is a corner of the polygon. In any polygon, the number of sides and vertices
are always equal.
The center is the point inside a regular polygon that is equidistant from each vertex.
The apothem of a regular polygon is the line from the center to the midpoint of a side.
The radius is the distance from the center to any vertex.
By definition, all sides are the same length, so the perimeter is simply the length of a
side times the number of sides.
Examples of regular polygons:
Area of a regular polygon:
8.- Circle:
A circle is a plane figure, bounded by a single curve line called its
circumference, every part of which is equally distant from a point
within it, called the center.
A circle sector is any piece of the circle between two radial lines
(shaded in both dark and clear grey).
A segment of a circle is the region between a chord of a circle
and its associated arc (shaded in dark grey).
An annulus is the region lying between two concentric circles
Areas
9.- Pythagorean Theorem:
Pythagorean Theorem:
In a right angled triangle, with sides (legs) a and b, and hypotenuse c, then
c² = a²+b².
The square of the hypotenuse is the sum of the squares of the legs.
A "3,4,5" triangle has a right angle in it, so the formula should work.
52=32+42
25=9+16
25=25
Yes, it works!!!
Apply the Pythagorean Theorem: If you know any two of the sides in a right triangle, you can
find the dimensions for the missing side using the Pythagorean Theorem.
a) Using the Pythagorean Theorem to find a missing leg
We can use the Pythagorean theorem to find a missing leg of a triangle, but only if we know the
length measure of the hypotenuse and the other one of the legs.
Find x
52=32+x2
x2=52-32
x2=25-9
x2=16
x=4
b) Using the Pythagorean Theorem to find the hypotenuse
We can use the Pythagorean theorem to find the hypotenuse of a right angled triangle, but only if
we know the length measure of the two legs.
Find x
c2=a2+b2
c2=82+122
c2=64+144
c2=208