Chapter 9
9.3 Altitude on Hypotenuse Theorem
9.5 Distance Formula and Coordinate Geometry
9.8 Space figures and the Pythagorean Theorem
10 ? multiple choice review on 9.6-8
9.9
Introduction to Trigonometry
Review for 9.7-9.10 (2003)
You must be able to simplify radicals.
A radical consists of 3 parts. The radical sign, the index and the radicand.
The index is 3, the radicand is 8 and the
radical sign is
The above example means you are looking for a number that when multiplied times itself 3 times will give you an answer of 8. In this case the answer is 2.
You will be taking square roots of numbers which means you are looking for a number times itself that will give you the radicand.
Often times you will not find a number that will give you the exact amount of your radicand so you must put your answer in simplest form. Below are the rules for simplifying radicals with examples of each.
Rules for Simplifying a radical.
1. No factor of the radicand may be a perfect square.
ex.
2. No fractions may be left as radicands.
ex.
3. No radicals may be left in the denominator. The process of eliminating a radical in the denominator is referred to a rationalizing the denominator.
ex.
4. An index must be as small as possible.
ex.
Rules for Operations on Radicals
1. To add or subtract radicals. The radicands must be the same. Keep the radicand and add or subtract the coefficients. Sometimes you must simplify the radicals before adding or subtracting.
ex.
2. To multiply or divide radicals. Multiply or divide coefficients, multiply or divide radicands, simplify if possible.
ex.
Solving Quadratic Equations
1. Set the equation = 0
2. Factor
3. Set each factor = 0
ex.
Some of you may have learned to factor by the multiplying up method.
Multiply the coeficient of the first term by the constant term
(18) (-20)= -360 Find factors of -360 that add to the middle terms coefficient of -9
(-24) (15) then write two binomials using the coefficient of the first term.
(18x-24) (18x+15) Since you have an extra 18 you must divide by factors of 18 to get rid of the extra. (18x-24)/6 (18x+15)/3 = (3x - 4) (6x + 5)
If there is no linear term ( x term) you can isolate the squared term on one side of the equation and take the square root of both sides. Just make sure to give both signs.
ex.
We have already defined a circle and learned the formulas for the area and circumference of a circle in a previous chapter but just to review.
Circle - definition - the set of all point in a plane a given distance from a given point
Symbol -
A radius of a circle is the distance from the center to any point on the circle
A chord is any segment that joins two points on a circle
A diameter is any chord of a circle that contains the center of the circle.
A diameter = 2 radii
The formula for the area of a circle :
The formula for the circumference of a circle:
New Material
Central Angle = an angle formed in a circle whose vertex is the center of the circle and whose sides are radii of the circle
An arc of a circle is the portion of a circle between two points, including these points.
A minor arc is an arc that is less than 1/2 of a circle. Its symbol is the two endpoints of the arc with a small eyebrow over the top as above.
A major arc is an arc that is greater than 1/2 of a circle. Its symbol is endpoint-point on arc- endpoint, with a larger eyebrow over the top. Below is an example of a major arc and its symbol.
The measure of an arc is defined to be = to the measure of the central angle that intercepts the arc.
The length of an arc is the portion of the circumference of the circle so its measure is one of length and not degree rotation. Be Careful as to whether the measure (degree) of an arc is being asked for or the length ( cm, inches etc)
To find the length of an arc use the formula below:
A sector of a circle is a region bounded by two radii and an arc of the circle.
The area of the sector is the portion of the area of the circle and is found using the formula below.
The last concept in this section is the method of finding the measure of an angle whose vertex is on the circle and whose sides are chords of the circle.
This angle is called an Inscribed Angle
We will prove in Chapter 10 that this angle is = 1/2 the measure of its intercepted arc.
For now just memorize it.
Note a central angle = measure of its intercepted arc
an inscribed angle = 1/2 measure of its intercepted arc.
9.3 Altitude to the Hypotenuse Theorems
In the last chapter we discovered that when the altitude was drawn to the hypotenuse of a right triangle it divides the triangle into 3 similar triangles.
When comparing the two small
triangles 
note that AD is a side of both.
When the proportions is set up between the two smallest sides and the two medium sides, AD gets used twice and becomes a geometric mean between the segments on the hypotenuse.
When the altitude is drawn to the hypotenuse of a right triangle it becomes the geometric mean between the segments on the hypotenuse.
When the small triangle is set proportional to the original triangle and proportions are set up, AC ( leg of the original triangle) is both a side of the small triangle and the original triangle. It becomes a geometric mean between the hypotenuse and the segment of the hypotenuse nearest the leg.
When the altitude is drawn to the hypotenuse of a right triangle each leg becomes a geometric mean between the whole hypotenuse and the segment of the hypotenuse nearest that leg.
example:
Given:
KF // HJ
1.
Find the coordinates of K and J
answer Find the slope from G to F
by 
This will give you -2/3 for
the slope. Since HJ and KF both have a slope of -2/3 you can use this with
the coordinates to find the missing x or y value.
H= (-1,4) J = (x,-2) remember points on horizontal lines have the same y value. F and I both have y values of -2
Do the same for K (-1,y) and F(5,-2)
2.
Find the length of HJ
Since side HI = 6 (subtract y values of H
and I) and IJ = 9 use 
to find the
length of HJ 
3.
4.
ans
ans. 
5. Find the perimeter of a
rectangle whose length is
and whose width is
in simplest form.
ans.
6.
Find the area of a rectangle whose length is
and whose width is
.
ans.
7.
8.
ans. (x-20)(x+5) = 0, x=20 or x = -5 (2x+3)(x-5)=0, x=-3/2 or x = 5
Given the diagram below, answer questions 9 - 11.
9. Find the measure of arc CD.
ans.
the arc is equal to the central angle 72 degrees
10. Find the length of arc CD
ans.
11. Find the area of sector CAD
12. Find
the measure of
.
ans. <CED = 90 as it is inscribed on a semi circle of 180 degrees. Inscribed angles measure 1/2 of their intercepted arcs.
13.
Find the measure of
ans. Arc ED = 112 degrees as it is what is left after you subtract 68 from the semi circle of 180 degrees.
Since
<ECD intercepts an arc of 112 degrees, it measures 56 degrees.
Find
the missing lengths in each of the following diagrams.
14.
x =
________15. If KM = 20, the length of GI.
16.
The sides of a triangle are 10, 15 and 18.
How long are the segments into
which the bisector of the largest angle separates the opposite side?
In a right triangle, the square of the hypotenuse is equal to the sum of the squares of the legs.
There are quite a few ways to prove the Pythagorean Theorem. The proof below is one of the methods of proving the theorem.
Pythagoras was a Greek man who lived in the 6th century BC. He is credited with first proving the above theorem although others have come up with subsequent proofs. President Garfield is credited with proving a theorm using a trapezoid as his base. Below is a site that demonstrates some of the many ways to prove the theorem.
http://www.geom.umn.edu/~demo5337/Group3/promenu.html
The following is one of the proofs of the theorem.
The following site is a calculator for finding a missing length of a right triangle. It also will give you the measures of the acute angles of your triangle which is a trigonometric function you will be learning soon.
http://www.1728.com/pythgorn.htm
For fun try this site which has an interactive proof of the Pythagorean Theorem.
http://www.sunsite.ubc.ca/LivingMathematics/V001N01/UBCExamples/Pythagoras/pythagoras.html
Remember the pythagorean theorem only works with right triangles
Examples.
Given the following find each missing side or segment asked for.
1. 2.
Sometimes you must drop a perpendicular to find the length you are asked for.
3. Given Isosceles triangle with base of 18 and legs of 15, find the height (altitude of the triangle)
The altitude of an isosceles triangle splits the base into 2 congruent segments. The right triangle has a hypotenuse of 15, a leg of 9 and we are looking for the second leg.
9.5 Distance Formula and Coordinate Geometry
To find the distance between two points on a Cartesian Coordinate Plane we use a formula based on the Pythagorean Theorem
Notice that it is easy to find the lengths
of horizontal and vertical segments on a grid as you need only take
the
absolute value of the subtraction of the x values of your coordinates to find
the horizontal segment
and
the absolute value of the subtraction of the y values to find the vertical segment.
The
diagonal segment is more difficult to find as you need to use the Pythagorean
Theorem as it is the hypotenuse of the triangle shown.
Since a
right triangle can be built on any diagonal segment with the segment being the
hypotenuse and the sides of the triangle will be the subtraction of the x and y
coordinates, we can derive the distance formula .
To
find the distance between two points (x1,y1) and (x2,y2)
A Pythagorean triple is a set of three whole numbers a, b, c with the property that
Memorize the following 4 sets of triples.
3,
4, 5
5,
12, 13
8,
15, 17
7,
24, 25
It is easy to find the third side of a right triangle that is a multiple or reduction of the above triples
Notice the above triangle has side lengths
that are both divisible by 9. The
resulting division yields a 4 and 5. The
triangle is therefore a multiple of 3-4-5
and the 3rd side is 9 x 3 or 27 units long.
***You can still find the length of the 3rd side using the Pythagorean theorem, but by knowing the triples you can easily come up with the answer without a calculator.
Here is a method found on the following website for generating Pythagorean triples. http://www.maths.uts.edu.au/numericon/triples.html
Take any two fractions whose product is 2.
Cross multiply to get integers a, b in the same ratio. |
39 and 80
Calculate
c =
to
find c
There
are two special right triangles that you need to memorize.
45-45-90 – If a triangle has angles of 45°-45°-90°, the two legs are congruent and the hypotenuse is equal to the product of the leg times the square root of 2.
Since a 45-45-90 must be an isosceles triangle, then the hypotenuse
could be calculated using the Pythagorean Theorem
No matter what the length of the side is, the
hypotenuse will always be
times greater if the triangle has
angles of 45-45-90
Rules:
If you know
the leg, multiply by
to get the hypotenuse
I
to find the leg.
Examples:
Problem 1
Multiply 9 by
to find the hyp: 9
Problem 2 Divide 8 by
to get 4
Problem
3: Multiply 7
by
to get 14
Problem 4: Divide 5
by
to get 5
30-60-90
Triangle –
Use
the Pythagorean theorem on the sides to solve for the missing segment
In
a 30-60-90 triangle the side opposite the 30°=
1/2
hypotenuse.
To get back to the short leg (opposite the 30) from the long leg (opposite the 60) divide by
.
Hyp =
Face - Each flat surface of a prism is
referred to as a face. The number of faces on a prism depends upon the
number of sides on its polygonal
A rectangular prism has 6 faces, a triangular prism has 5 faces, the
hexagonal prism has 8 faces.
Edges - The intersection of two faces of a
prism forms a line segment which is referred to as an edge. In the prism shown
below, BE is referred to as an edge. There are 12 edges on the rectangular
prism. You can count them or multiply 3 times the number of sides on the
base.
Vertices - The point where 3 faces intersect is referred to as a vertex of
the prism. Point B is a vertex of the prism in the picture above.
There are 8 vertices on a rectangular prism. To find the number of
vertices, you can count them or multiply 2 times the number of sides on the
base.
To find the diagonal across a rectangular
prism find the square root of the sum of the squares of the prisms dimensions.
Pyramids - A pyramid is a 3 dimensional figure with a polygonal base and
sides that are triangles that intersect at one point. Below are pictures of
pyramids.
Below are pictures of 3 different pyramids.
The number of faces on each is determined by the number of sides on the base
Triangular pyramid - 4 faces (one more than the number of sides)
Square pyramid - 5 faces
Hexagonal pyramid - 7
The number of vertices (points where three faces intersect) is the same as the number of faces.
Vertices: Triangular pyramid = 4 Square pyramid
=5 Hexagonal pyramid = 7
The number of edges (segments where two faces intersect) = twice the number of sides on the base.
Triangular pyramid has 6 edges ( 3 x 2)
Square pyramid has 8 edges
Hexagonal pyramid =12
The slant height ( l ) of a pyramid is the altitude of each triangle. Since in a regular pyramid the triangles are all congruent, the slant heights are the same.
Most problems on the surface area of a pyramid have you solve for the length of the slant height by using the pythagorean theorem. Below are two examples of solving for the slant height.
Ex. 1
Find the length of the slant height given that the base edge = 24 and the lateral edge = 15
Ans. The perpendicular splits the base edge into two congruent segments of 12 each. Using the pythagorean theorem 152=122+ l2 , l = 9
Ex. 2
Find the length of the slant height given the base edge = 12 and the height of the pyramid =8
Ans. The right triangle formed by the height of the pyramid, the slant height and the segment from the center of the square to the side is what will be used in the Pythagorean theorem. The length of the segment from the center to the side is 6. The equation will be l2 =82+ 62 , l = 10