Honors - Proportions and Similarity
Name:
Section 1: Proportions
Define:

Extremes:

Geometric Mean:

Mean Proportion:

Mean Proportional:

Means:

Proportion:

Ratio:
Theorems:

Means-Extremes Products Theorem (MEPT): in proportion, product of means is equal to product of
extremes. (if
then ad = bc)
Examples: Write a Ratio
1. There are 76 boys in the freshmen class of 165 students. Find the ratio of boys to girls.
Total – Boys = Girls
165 – 76 = 89
Ratio of Boys to Girls is 76:89
2. A hockey player scored 4 goals in 12 games. Find the ratio of goals to games.
4:12 = 1:3
Example: Ratios in Triangles
3. In a triangle, the ratio of the measures of the three sides is 4:6:9 and its perimeter is 190 inches. Find the
length of the longest side of the triangle.
4x + 6x + 9x = 190;
19x = 190;
x = 10
Longest side: 9(10) = 90
Practice: Ratios in Triangles
4. The ratio of the measures of the three sides of triangle is 8:7:5. Its perimeter is 240 feet. Find the measure
of each side.
5. The ratio of the measures of the three sides of triangle is 3:4:5. Its perimeter is 72 inches. Find the
measure of each side.
Proportions & Similarity Notes
1
Rev E
Honors - Proportions and Similarity
Examples: Solving Proportions
6. If
Name:
what does x equal?
7. If
Find the cross products and then solve for x.
3(14) = 7x
42 = 7x
6 =x
what does x equal?
Find the cross products and then solve for x.
15(2x + 1) = 35(3)
30x + 15 = 105 (Distributive Property)
30x = 90 (Subtraction Prop)
x = 3 (Division Prop)
8. Find the 4th term (4th proportional of proportion with 1st 3 terms 2, 3, 4).
Set up the proportion such that the x is in the 4th term. Find the cross product and then solve for x.
2x = 12; x = 6
9. Find the mean proportional(s) between 4 and 16.
Mean proportional has the same mean value; therefore, set it up with x’s in both mean positions, find the
cross product and solve for x.
This is the same as saying x2 = 4(16) = 64; x =  8
10. If 3x = 4y, find the ratio of x to y.
Need in the form of
so need to manipulate above equation to get it in that form. Result:
11. A twinjet airplane has length of 78m and wingspan of 90m. A toy model is made in proportion to real
airplane. If wingspan of toy is 36m, find the length of toy.
(78)(36) = 90x;
x = 31.2
Hint: always put information of the 1 object on top and information from 2nd object on the bottom.
Practice: Proportions:
12. If
what does x equal?
13. If
what does x equal?
14. Find the mean proportional(s) between 4 and 25.
15. If 2x = 3y, find the ratio of y to x.
16. Find the 3rd term (3rd proportional of proportion with 3 terms 1, 2, 8.
17. A twinjet airplane has length of 80m and wingspan of 100m. A toy model is made in proportion to real
airplane. If wingspan of toy is 25m, find the length of toy.
18. The scale drawing of a family room is 3in by 5in. If the scale is 1in:6ft, what are the actual dimensions of
the family room?
Proportions & Similarity Notes
2
Rev E
Honors - Proportions and Similarity
Section 2: Similar Polygons
Define:




Name:
Dilation:
Scale Factor:
Similar (~):
Similar Polygon:
Theorems:

Perimeter Ratio Theorem: The ratio of the perimeters of 2 similar polygons equals the ratio of any pair of
corresponding sides.
Example: Jacob is making a scale model of the White House. The perimeter of the White House is 506 feet.
Jacob will use a scale factor of 5feet to 1 inch. What will the perimeter of the model be?
;
x = 101.2in
Examples/Practice: Similar Polygon
Determine whether each pair of figures is similar. Justify your answer.
19.
20.
Example: Proportional Parts and Scale Factor
21. Two polygons are similar. Write a similarity statement. Then find x and y and the scale factor of polygon.
Examples/Practice: Enlargement of Figure
22. ∆ABC is similar to ∆XYZ with a scale factor of . If lengths of sides of ∆ABC are 6, 8, 10, what are
lengths of sides of ∆XYZ?
23. Jackie used a scale factor of to enlarge a photograph. The original dimensions of the photo are 6cm by
9cm.
a. Find the dimensions of the new photo.
b. Find the perimeter of the new photo.
Proportions & Similarity Notes
3
Rev E
Honors - Proportions and Similarity
Section 3: Similar Triangles
Theorems:

AA~ Theorem: if 2
of 1 ∆ are
to 2
of other ∆, then ∆s are ~.
C
H and E
Name:
J
∆HIJ ~ ∆CDE

SSS~ Theorem: if measures of corresponding sides of 2 ∆s are proportional then ∆s are ~.
∆XYZ ~ ∆PNM

SAS~ Theorem: if measures of 2 sides of ∆ are proportional to measures of 2 corresponding sides of
another ∆ and included angles are  , then ∆s are ~.
P
J,
∆JKL

∆PRQ
Misc
Similarity of triangles is reflexive, symmetric and transitive
o Reflexive: ∆ABC ~ ∆ABC
o Symmetric: ∆ABC ~ ∆DEF  ∆DEF ~ ∆ABC
o Transitive: ∆ABC ~ ∆DEF and ∆DEF ~ ∆GHI  ∆ABC ~ ∆GHI
Examples/Practice
24. The sides of 1 triangle are 8, 14, 12 and the sides of another triangle are 18, 21 and 12. Are the 2 triangles
similar or not?
25. The sides of 1 triangle are 2.4, 3.6 and 4.8 and the sides of another triangle are 8, 12 and 15. Are the 2
triangles similar or not?
26. While taking a walk one evening, I noticed that a 20-m flagpole cast a 25-m shadow. Nearby I saw a
telephone pole that cast a 35-m shadow. How tall was the telephone pole?
27. Frank stands so that his shadow and that of the flagpole are in line and have the same tip. Frank’s height
is 178cm and his shadow is 267cm long when the pole’s shadow is 921cm long. How tall is the flagpole?
Proportions & Similarity Notes
4
Rev E
Honors - Proportions and Similarity
Name:
Section 4: Parallel Lines & Proportional Parts
Theorems:

Triangle Proportionality/Side Splitter Theorem: if a line is || to 1 side of ∆ and intersects other 2 sides in
2 distinct points, then it separates these sides into proportional segments.
if BE || CD,
Example:
If DE = 9, AE = 21 and BC = 6, what is AB?
→
→9x = 6(21) → x = 14

Converse of Side Splitter Theorem: If line intersects 2 sides of ∆ and separates sides into corresponding
proportional segments, then line is || to 3rd side.
If

then BE || CD
Triangle Midsegment Theorem: If a segment joins the midpoints of two sides of a triangle, then it is
parallel to the third side and its measure equals one-half the measure of the third side.
If B and E are midpoints of AC and AD, then BE || CD
and BE = ½CD

Three Parallel Lines Corollary: if 3 or more parallel lines are intersected by 2 transversals, the parallel
lines divide the transversals proportionally.
If AB || CD || EF, then
Practice:
28. Given the diagrams below, complete the following statements:
b.
a.
c.
29. AC = 15, AB = 10 and AE is twice the length of ED. Determine
whether BE || CD. Explain.
30. AD = 8, AE = 3, AB = 6, find BC.
Proportions & Similarity Notes
5
Rev E
Honors - Proportions and Similarity
31. Find EF
Name:
32. Find the missing length.
33. Find the missing length.
34. ∆ABC has vertices A(-2,9), B(-4,1) and C(8,-1). DE
is mid-segment of ∆ABC.
a. Find coordinates of D and E (Hint: Midpoint
formula)
b. Verify that BC is parallel to DE. (Hint: slopes
must be same)
c. Verify that DE = ½BC (Hint: Distance formula)
y
10
9
8
7
6
5
4
3
2
1
–9 –8 –7 –6 –5 –4 –3 –2 –1–1
–2
–3
–4
–5
–6
–7
–8
–9
–10
35. ∆ABC has vertices A (-1,6), B(-4,-3) and C(7,-5).
DE is mid-segment of ∆ABC.
a. Find coordinates of D and E
b. Verify that BC is parallel to DE.
36. Given: AB = 3x-4, BC = 6-2x, DE =
AB  BC. Find x and y
Proportions & Similarity Notes
6
3
4
5
6
7
8
9 x
1
2
3
4
5
6
7
8
9 x
y
and EF = 3y. If
37. Given: AB=5, BC=2, CG=8. If DH = 24, find DE.
2
10
9
8
7
6
5
4
3
2
1
–9 –8 –7 –6 –5 –4 –3 –2 –1–1
–2
–3
–4
–5
–6
–7
–8
–9
–10
c. Verify that DE = ½BC
1
AB || BE|| CF
AD || BE || CF|| GH
Rev E
Honors - Proportions and Similarity
Name:
Section 5: Parts of Similar Triangles
Theorems:

Proportional Perimeters Theorem: if 2 triangles are ~, then their perimeters are proportional to the
measures of the corresponding sides.
If ∆ABC ~ ∆DEF then

Angle Bisector Theorem: if a ray bisects an angle of triangle, it divides opposite side into segments that
are proportional to adjacent sides.
If AC bisects
then
Special Segments of Similar Triangles

Altitude Proportional Theorem: ~ ∆ have corresponding altitudes proportional to corresponding sides.

Median Proportional Theorem: ~ ∆ have corresponding medians proportional to corresponding sides.

Angle Bisector Proportional Theorem: ~ triangles have corresponding angle bisectors proportional to
corresponding sides.
Examples/Practice:
38. If ∆LMN ~ ∆QRS, find the perimeter of ∆LMN.
39. If 2 similar triangles have perimeter 21 and 28, what is the ratio of the measures of the 2 corresponding
sides?
40. ∆MNP ~ ∆RST, the perimeter of ∆RST is 9 units, NM = 3, MP = 4.5 and NP = 6. Find the value of each
side of ∆RST.
41. Suppose ∆DEF ~ ∆GHI and the scale factor of ∆DEF to ∆GHI is . Find the perimeter of ∆GHI if the
perimeter of ∆DEF is 24m.
Proportions & Similarity Notes
7
Rev E
Honors - Proportions and Similarity
Name:
42. The perimeter of a triangle is 88 units. The sides have length d, e and f. The ratio of e to d is 3:1 and the
ratio of f to e is 10:9. Find the lengths of each side.
43. For the following pair of similar triangles, find the values of a, b and c when the perimeter of ∆MON=36
20
15
10
44. ∆ABC ~ ∆DEF. Find EH if BC=30, BG=15, EF=15.
45. ∆ABC ~ ∆DEF. Find BG if BC=6.5, EF=13, EH=12.
Find the value of x.
46.
47.
48.
Proportions & Similarity Notes
49.
8
Rev E