- 5 -
Name:______Period:______
Geometry Rules! Chapter 7 Notes
Notes #36: Solving Ratios and Proportions and Similar Triangles (Sections 7.1 and 7.2)
Ratio: a comparison of two quantities. , 3 to 4, 3:4
Proportion: two ratios that are equal to each other.
A. Solving Proportions
· Cross-multiply and set the products equal to each other
· Be sure to use FOIL when you are multiplying binomials together
· Solve for the variable
Solve for x:
1.) / 2.)3.) / 4.)
B. Properties of Proportions
Complete:
5.) / 6.)7.) / 8.)
9.) / 10.)
11.) For the given figure, it is given that: . Solve for the missing lengths.
KR = 6, KT = 10, KS = 8/ RT = ____
SU = ____
KU = ____
C. Similar Polygons
Similar polygons have the same ______but not necessarily the same ______.Example of similar triangles:
· Their corresponding angles are ______
· Their corresponding sides are in a ______
· This ratio is called a ______and in this case is ______
· We show that they are similar with this statement: ______
12.)
/ a) scale factor = ______/ b),
c) x = _____, y = ______, z = ______
The figures are similar. Solve for the variables. (Hint: redraw the diagram as two figures)
13.)14.)
D. Algebra Practice:
Factor:
15.) / 16.) / 17.)Simplify:
18.) / 19.) / 20.)
Solve:
21.) / 22.) / 23.)
Notes #37: Similar Triangles (Sections 7.3 and 7.5)
Similar triangles have:
· ______corresponding angles
· sides that are in ______
You can conclude that two triangles are similar if:
______: two pairs of corresponding angles are congruent______: all three pairs of sides are in the same proportion
______: two pairs of sides are the same proportion and their included angles are congruent
Are the triangles similar? If so, state the similarity and the postulate you used.
· Re-draw the triangles in matching positions
· Mark congruent angles
· Test sides for a constant proportion:
· Look for these patterns: AA~, SSS~, SAS~
1.)/ 2.)
3.)
/ 4.)
5.)
/ 6.)
State whether the figures are always, sometimes, or never similar:
· do they always, sometimes, or never have the exact same shape?
7.) two squares 8.) two congruent triangles
9.) two rectangles 10.) two rhombuses
11.) two pentagons 12.) two regular octagons
Proportional Lengths (Section 7.5)
A. Triangle Proportionality
· A parallel slice cuts a triangle’s sides proportionally ( Side-Splitter Theorem)
Example: Solve for x
B. Angle Bisector Proportionality
· An angle bisector proportionally divides the opposite side
Example: Solve for x:
C. Parallel Line Proportionality
· Parallel lines proportionally divide their transversals
Example: Solve for x:
Solve for the variables:
/ 2.)
3.)
/ 4.)
5.)
/ 6.)
13.) Write the equation of a line that contains the point ( -4, 3) and has a slope of . / 15.) Write the equation of a line in standard form that is parallel to and contains the point ( 1, 4).
16.) Write the equation of a line that contains ( -2, -3) and ( 4, -9) in standard form. / 18.) Write the equation of a line that is perpendicular to and contains the point ( -6, 7)
Notes #38: Similarity in Right triangles ( 7.4)
Geometric Means and Similar Right TrianglesA. Geometric Mean
asks the question: “what number, squared, equals the product of two given numbers?”
Find the geometric mean of the listed numbers:
· Use the given numbers in this equation: x2 = ab
· Solve for x
1.) 9 and 16 / 2.) 12 and 3 / 3.) 5 and 15
B. Similar Right Triangles
· When an altitude of a right triangle is drawn to its hypotenuse, three similar right triangles are formed:
Solve for the variables:
· Re-draw the three triangles and label all sides
· Set up proportions to solve for the variables
· Look for ways to use the Pythagorean theorem
4.)
/ / /
5.)
6.)
Notes #39—Section 8.1
Pythagorean Theorem
In Words:In a ______triangle, the sum of the ______of the lengths of the ______is equal to the ______of the length of the ______. / Pictures/Symbols: / Example:
Find the missing side of the triangles below.
1.)
2.)
A is a set of whole numbers a, b, and c, that satisfy the Pythagorean Theorem.
Examples: Do the lengths of the sides given form a Pythagorean triple?
3.) 8, 15, 17 4.) 7, 4, 6 5.) 20, 21, 29
Examples: Find the value of x. Leave your answer in simplest radical form.
6.) 7.)
8.)
Determining Whether a Triangle is Right, Acute, or Obtuse Given Three Side Lengths:
RightEx 9: Sides have lengths 3, 4, and 5 / Acute
Ex 10: Sides have lengths 12, 6, and 11 / Obtuse
Ex 11: Sides have lengths 14, 7 and 12
Examples: The lengths of the sides of a triangle are given. Classify the triangle as acute, right, or obtuse.
12.) 10, 15, 20 13.) 7, 6, 4 14.) 15, 20, 25
Examples: Find the value of x. Leave your answer in simplest radical form.
15.) 16.)
Algebra Review: Solve using quadratic formula
Examples:
17.) 18.)
19.) 20.)
Notes #40: Chapter 7 Review
Simplify each ratio:
1.) a) BC:CD b) m<B:m<Cc) CD: Perimeter of ABCD /
2.) If x = 4, y = 6, z = 2 find each ratio:
a) x to y b) (x + z) to y c)
3.) / 4.)
Write and equation and solve:
4.) The ratio of the angles of a triangle is 1:3:5. Find the angles.
5.) The ratio of the angles of a pentagon is 6: 8: 9: 11: 11. Find the angles.
Are the triangles similar? If so, write a similarity statement and the postulate you used:
/ 7.)
8.)
/ 9.)
Solve for the variables:
10.) / 11.)/ 12.)
13.)
Simplify: / 14.)
15.) Similar Right Triangles: Solve for m, n, and p in reduced radical form.
/ 16.) a.) Find the geometric mean of 5 and 10
b.) Find the geometric mean of 4 and 20.
Are the figures sometimes, always, or never similar?
17.) two rectangles 18.) two equilateral triangles 19.) two regular hexagons
Study Guide 7 Name:______
Show all your work! Date:______Period:______
For #1-3, ABCD is a parallelogram. Simplify each ratio:
1.) BC:CD2.) AD:(Perimeter of ABCD)
3.) / / 1.) ______
2.) ______
3.) ______
For #4 – 6, complete each statement:
4.) If a : 3 = 7 : 4, then
4a = _____ / 5.) If , then / 6.) If , then
/ 4.) ______
5.) ______
6.) ______
For #7-10, solve for x: / 7.) x = ______
8.) x = ______
9.) x = ______
10.) x = ______
7.) / 8.)
9.) / 10.)
For #11 – 16, state whether the two polygons are always, sometimes, or never similar. / 11.) ______
11.) two right triangles
13.) two squares
15.) an isosceles triangle and a right triangle / 12.) two scalene triangles
14.) two rectangles
16.) two regular hexagons / 12.) ______
13.) ______
14.) ______
15.) ______
16.) ______
For #17 - 20, refer to the diagram.
17.) Find
18.) Find the scale factor of MATH to M’A’T’H’
19.) Solve for x.
20.) Solve for y. /
MATH ~ M’A’T’H’ / 17.) ______
18.) ______
19.) x = ______
20.) y = ______
For #21 – 34, complete the similarity statement and state why the triangles are similar. If the triangles are not similar, circle not similar. (If you are using SAS similarity or SSS similarity, be sure to check your side lengths for a common proportion) / 21.)
by ______
OR
not similar
22.)
by ______
OR
not similar
21.)
/ 22.)
23.)
/ 24.)
/ 23.)
by ______
OR
not similar
24.)
by ______
OR
not similar
For #25 – 28, solve for x and y (where x and y are positive):
25.)
/ 26.)
/ 25.) x = ______
y = ______
26.) x = ______
27.) x = ______
28.) x = ______
29.) ______
30.) ______
31.) ______
32.) ______
27.)
Solve: / 28.)
29.)
/ 30.)
/ 31.)
/ 32.)
For #33-34, find the geometric mean of the two numbers.
33.) 5 and 10 / 34.) 4 and 20 / 33.) ______
34.) ______
For #35-36, solve for x, y, and z. (Hint: use 3 similar, right triangles)
35.)
/ 35.) x = ______
y = ______
z = ______
36.)
/ 36.) x = ______
y = ______
z = ______
For # 37-38, solve for x. Leave the answer in simplified radical form
37)
/ 38.)
/ 37.)
x = ______
38.)
x = ______
For #39-40, the lengths of the sides of a triangle are given. Classify each triangle as acute, right, or obtuse.
39.) 20, 30, 40 / 40.) 41, 9, 40 / 39.)______
40.)______