Download Geometry Final Exam Review Materials

Geometry
Final Exam
Review Materials
Prepared by
Francis Kisner
June 2015
Proportional Reasoning
If 12 eggs are enough for one recipe that will feed 8 people, how many
eggs will be needed to serve 20?
Congruent line segments are the same length. Congruent angles are the
same measure.
Comparing two polygons that look the same shape, if all the sides of one
are the same multiple of the sides of the other, the sides are said to be
proportional.
What do you have if the angles of the two polygons are congruent and the
corresponding sides are proportional?
What does the symbol “~” mean in geometry?
If you have two similar polygons and you know the perimeter of one and
you know the “scale factor”, can you find the perimeter of the other?
Example: You have two trapezoids that are similar. One is ¾ the size of
the other. The larger has sides measuring 6, 2, 4, 4 inches. Can you tell
the perimeter of the smaller trapezoid?
Conditions giving similar triangles.
AA, SAS, SSS Pages 381-395
In triangles, knowing that the angles of two triangles are the same is
enough to show that the triangles are similar. If corresponding sides are
equal then the triangles are congruent.
When two lines cross they create two sets of angles. Within each set, the
angles have the same measure. ( Vertical Angles )
If a line (a transversal) is drawn across two parallel lines, the
corresponding angles formed are equal. (Literally, all the acute angles are
the same and all the obtuse angles are the same.)
The Triangle Proportionality Theorem and Converse
See pages 397 and 398
If a line parallel to one side of a triangle intersects the other two sides, it
divides the two sides proportionally.
If a line divides two sides of a triangle proportionately, then it is parallel to
the third side.
--If three parallel lines intersect two transversals, they divide the transversals
proportionately.
If a ray bisects an angle of a triangle, it divides the opposite side into
segments whose lengths are proportional to the lengths of the other two
sides. See page 398
Pythagorean Theorem
𝐼𝑛 𝑎 𝑟𝑖𝑔ℎ𝑡 𝑡𝑟𝑖𝑎𝑛𝑔𝑙𝑒 𝑤𝑖𝑡ℎ 𝑙𝑒𝑔𝑠 𝑎 𝑎𝑛𝑑 𝑏 𝑎𝑛𝑑 ℎ𝑦𝑝𝑜𝑡𝑒𝑛𝑢𝑠𝑒 𝑐,
𝑎2 + 𝑏 2 = 𝑐 2 𝑠𝑜 𝑐 = 𝑎2 + 𝑏^2
When will you need to use this?
Every time a problem involves two sides being at right angles to each
other, look to see if there is a right triangle being formed by some other
line.
If you are given the legs, you can find the hypotenuse.
If you are given one leg and the hypotenuse, you can find the other leg.
𝑎 = (𝑐 2 − 𝑏 2 )
Always remember that you have to take the square root to get the length.
Know the definitions of the Trig Functions
In a right triangle, when you are looking at an angle between a leg and the
hypotenuse:
Sin = opposite leg / hypotenuse
Cos = adjacent leg / hypotenuse
Tan = opposite leg / adjacent leg
Know how to find the functions on your calculator.
Given the value of a function, know how to find the angle.
Acute or Obtuse ?
If 𝑐 2 > 𝑎2 + 𝑏 2 then the triangle is obtuse.
If 𝑐 2 < 𝑎2 + 𝑏 2 then the triangle is acute.
Know how to find the sum of the interior angles of a polygon.
How does the sum relate to the number of sides?
Know the sum of the exterior angles of a polygon. Does the sum relate to
the number of sides?
Special Quadrilaterals
Know the definitions and characteristics of
Square
Rectangle
Parallelogram
Rhombus
Trapezoid
Kite
Which have parallel sides? How many parallel sides do they have?
Is there any special characteristic of the diagonals?
What do you know about parallelograms?
Sides?
Opposite Angles
Opposite Sides
Diagonals?
Consecutive angles
What do you know about trapezoids?
Sides?
Angles
Opposite Sides?
If we know the lengths of the bases, can we find the length of a line
connecting the centers of the other two sides?
What do you know about isosceles trapezoids?
Base angles?
Opposite sides? Diagonals?
Base segments?
Sides?
What do you know about kites?
Diagonals?
Opposite Sides?
Opposite Angles?
Interior Angles?
Consecutive Side Pairs?
What do you know about rectangles?
Sides?
Angles?
Diagonals?
Circles, Tangents, Chords, Etc.
If the tangent is 20cm long and the line from the center A to the top vertex
is 25 cm long, how long is the radius of the circle?
25
20
A
If mPR = 90º and PQ = QR, find mQR
P
R
Q
What do you know about triangles inscribed in circles?
What do you know if one side of the inscribed triangle is a diameter of the
circle?
Tangents and Chords
P
If the angle between the chord PQ
and the tangent line is 80º,
What is the measure of arc PQ ?
Q
150º
P
Q
100º
S R
If the angle between the chords is 100º and the arc PA measures 150º, what
is the size of arc RS?
Do you know about the proportions of intersecting chords?
If you know the lengths of some of the segments, can you find the others?
If you know some of the lengths of segments of Secant Lines, can you find
the others?
If you know the measure of arc ADC is 170º, can you find the measure of
A
angle ADC?
B
D
C
What do you know about the relationship between the radius of a circle
and a chord of the circle?
Do you know any other
facts about the chord and
the radius?
Can you find the area of a rectangle?
Can you find the height of a triangle if you know the base and the area?
Can you find the area of a parallelogram if you have the base and the
height?
Can you find the area of a kite if you have the lengths of the two diagonals?
If two polygons are similar and you know the ratio of their corresponding
sides, can you find the ratio of their areas?
For example, we have two hexagons and one had sides 9 feet long while
the other had sides 20 feet long. We know the area of the smaller hexagon
is is 93 sq.ft. To the nearest tenth of a square foot, what is the area of the
larger hexagon?
Solution:
The areas of similar figures are related to the lengths by this ratio: If the
sides of the figures are a/b then the areas are a2/b2
𝐼𝑓 𝑐𝑜𝑟𝑟𝑒𝑠𝑝𝑜𝑛𝑑𝑖𝑛𝑔 𝑠𝑖𝑑𝑒𝑠 𝑎𝑟𝑒 𝑎 𝑎𝑛𝑑 𝑏 𝑡ℎ𝑒𝑛
𝐴𝑟𝑒𝑎 𝑜𝑓 𝐴 𝑎2
= 2
𝐴𝑟𝑒𝑎 𝑜𝑓 𝐵 𝑏
So
𝐴𝑟𝑒𝑎 𝑜𝑓 𝑙𝑎𝑟𝑔𝑒𝑟
93
=
202
92
=
400
81
→ Area = 459.3 ft 2