Transcript Section 10.3 - Mathematical sciences

Section 10.3
Angles and Phenomena in the World
Reflections
• Def: A normal line at a point on a surface is the line that
passes through that point and is perpendicular to the surface
at that point.
• Law of Reflection: Incoming and reflected light rays make the
same angle with the normal line at the point where the
incoming light ray hits the surface.
Section 10.4
Circles and Spheres
Definition of a Circle
• Def: A circle is the collection of all the points in a plane that
are a fixed distance away from a certain point.
• Def’s: The fixed point is the center and the distance is the
radius. The diameter is the distance from a point on the
circle to the point “opposite” of it.
Example Problem
• Ex 1: Lexington and Louisville are 75 miles apart. Stephanie
lives 30 miles from Lexington and 90 miles from Louisville. Is
it possible to know exactly where Stephanie lives from this
information?
Definition of a Sphere
• Def: A sphere is the collection of all points in space that are a
fixed distance away from a certain point.
• Def’s: Center, radius, and diameter are defined as before.
Section 10.5
Polygons
Plane Shapes
• Def: A plane shape or 2-dimensional shape (or shape) is a flat region
that lies in the plane that is connected and closed.
• If the shape is made of line segments, every endpoint of a segment
must meet at exactly one endpoint of another segment, and
segments are not allowed to cross.
4-sided Shapes
• Def: A quadrilateral is a closed shape in a plane consisting of 4 line
segments that do not cross
• What properties of shapes help us categorize or classify them?
Number of sides
o If any sides are parallel
o If any side lengths are equal
o The size of the angles
o Symmetry?
o
Classification of Quadrilaterals
• Square- quadrilateral with 4 right angles whose sides all have the same
length
• Rectangle- quadrilateral with 4 right angles
• Rhombus (Diamond)- quadrilateral whose sides all have the same length
• Parallelogram- quadrilateral for which opposite sides are parallel
• Trapezoid- quadrilateral for which at least one pair of opposite sides are
parallel
3-sided Shapes
• Def: A triangle is a closed plane shape made of 3 line segments.
• Special Triangles:
o A right triangle is a triangle that has a right angle. The side opposite of
the right angle is the hypotenuse.
o A triangle with 3 sides of the same length is an equilateral triangle.
o A triangle with at least 2 sides of the same length is an isosceles
triangle.
o A triangle whose angles are all smaller than 90˚ is an acute triangle, and
one with an angle greater than 90˚ is an obtuse triangle.
Identifying Triangles
• Ex: Identify whether each of the following are right, equilateral,
isosceles, acute, and/or obtuse triangles.
Polygons
• Def: A polygon is a plane shape consisting of a finite number of line
segments.
𝑛-gon: polygon with 𝑛 sides and vertices
o 3-gon: triangle
o 4-gon: quadrilateral
o 5-gon: pentagon
o 6-gon: hexagon
o 8-gon: octagon
o 12-gon: dodecagon
o
• Def: A polygon is regular if all sides have the same length and all angles are
equal.
• Theorem: The interior angles of the regular 𝑛-gon are
180
360
−
𝑛
Venn Diagrams
• Venn Diagrams: use ovals to show how certain sets, or collections of
objects, are related