Geometry/Chapter 7

Section 7.1 - Parallel Lines

In Euclidean Geometry, lines are parallel if they do not intersect.
Remember that lines extend infinitely, if they are not parallel they will eventually intersect.

Exercises

draw line AB, draw a line CD that is parallel to AB
find a way to prove that AB is parallel to CD

Section 7.2 - Perpendicular Lines

In Euclidean Geometry, lines are perpendicular if they intersect at a 90 degree angle.

Exercises

Draw a line AB, then pick a point C on line AB, make a line which intersects C and is perpendicular to AB, i.e.,

$CD\perp AB$

prove that line CD is perpendicular to AB.

Section 7.3 - Circles

Circles

Congruent Circles

Congruent circles are circles that have the same size, that is, the same radius.

Concentric Circles

Concentric Circles share the same center, axis or origin with one inside the other. Concentric Circles do not necessarily have the same radius.

Exercises

draw a circle, find a way to draw a congruent circle, prove they are congruent
draw a circle, then draw a concentric one, prove that they are concentric.

Radius, Diameter, Chord

A radius is a line segment connecting a point on the circle and the center of the circle.

Diameter is a line segment connecting 2 points on the circle and goes through the center.

A chord is a line segment connecting 2 points on the circle that does not have to go through the center. The diameter of a circle is the longest chord.

Section 7.4 - Quadrilaterals

A quadrilateral is defined as any 4 sided polygon. This means squares, rectangles, etc. are all called quadrilaterals. In the following sections we will take a closer look at each of the important quadrilaterals, how they are defined, and some special properties of each of them.

Rectangles

A Rectangle is defined by having 4 conditions. The first is that it is a 4 sided polygon. The second is that it has 2 pairs of parallel sides. The third is that the pairs of parallel sides are of equal length. The forth is that all angles must be equal to 90 degrees or a right angle.

Lets take an example. Below is a Rectangle. First we note that it has 4 sides, and that they form a polygon, this makes it a 4 sided polygon so the first condition is satisfied. Next we notice that 2 of the sides are parallel to each other and that the other 2 sides are parallel to each other, thus the second condition has been meet. Now we notice that the pairs of parallel sides are of equal length, thus the third condition has been meet. The final condition was that all angles must be 90 degrees, as shown in the image they are in fact 90 degrees. Because all 4 conditions have been meet we now know that the polygon below is in fact a Rectangle.

Special Properties

Ok, now that the rectangle has been defined we need to know whats so special about rectangles.

Length of a diagonal:

Below is the same rectangle as above except it has the diagonal drawn from 1 vertex to the other. The length of this diagonal is equal to the square root of one side squared added to the length of the adjacent side squared. e.g. ${\sqrt {(}}3^{2}+1^{2})={\sqrt {(}}10)=3.162$

The proof of this is quite simple. But first you have to know the pythagorean theorem so read over this if you don't already know it.

1) Angle ABC = 90 degrees - given in the definition of a rectangle.
2) Side AB and side BC and the diagonal form a triangle. - a 3 sided polygon
3) The triangle is also a right angled triangle because it has a right angle in it. - angle ABC
4) The diagonal of the rectangle is the hypotenuse of the triangle.
5) the hypotenuse of a right angled triangle is equal to the square root of the sum of the squares of the sides - Pythagora's theorem
6) thus the diagonal is equal to ${\sqrt {(}}b^{2}+c^{2})$

Following this property we can note that if we draw a diagonal we create 2 congruent triangles inside the rectangle.

proof:

below is such a Rectangle with the diagonal drawn. We shall name the vertices A, B, C, D. The diagonal is called AC because the 2 endpoints are A and C. The 2 triangles are ABC and ADC.

1) The 2 triangles share the a side - AC.
2) AD and BC have an equal length - given.
3) angles ABC and ADC are of equal measure - definition of a rectangle.
4) Each triangle has an angle of equal measure - statement 3.
5) by the side angle side theorem each triangle is congruent.

Examples

Square

A square has the same properties as a rectangle except that all 4 sides must be equal length. A square is also considered a rhombus, kite, parallelogram, and trapezoid.

Special Properties

Examples

Trapezoid

A Trapezoid is a quadrilateral with 2 sides that are parallel.

Special Properties

Area for a trapezoid is given by the formula:

$A=h{\frac {a+b}{2}}.$ , where a and b are the lengths of the 2 parallel sides and h is the height.

Examples

<1=

Rhombus

A Rhombus is a quadrilateral with four equal sides.

Special Properties

The area of a Rhombus is given by the formula:

$Area={\frac {D_{1}\times D_{2}}{2}}$
where D₁ and D₂ are the lengths of the two diagonals

Because the rhombus is a parallelogram, the area also equals the length of a side (B) multiplied by the perpendicular distance between two opposite sides(H)

$Area=B\times H$