SIMILARITY in 2 DIMENSIONS

Similar or Congruent: What's the Difference?

There are two kinds of human twins: dizygotic or fraternal twins (from 2 eggs) and monozygotic or identical twins (from 1 split egg). Fraternal twins look alike but it's easy to tell them apart -- because usually one is a boy and the other a girl. Identical twins are identical. They look exactly alike and are always the same gender. Sometimes, even their mother can't tell them apart.

In geometry, when we say figures are similar, we mean they look alike, but they are not identical – like fraternal twins, it's easy to see the difference between them. With similar figures, the shape is the same, however, the size is different. We could place one inside the other and line them up so that their corresponding sides are parallel.

Two congruent figures are exactly the same -- like identical twins. There's no way to tell them apart because they are identical. Congruent means "equal in all respects" so, everything – the sides, the angles, the perimeter, and the area are exactly the same. We could place one figure over the other and they would match up perfectly.

Similar Figures

The graphics software we use today to "draw" images with our computers, makes it extremely easy to create similar and/or congruent figures. To make the 3 blue triangles in the diagram above, I " drew" one of them, then I selected it, did a " copy and paste", to make the 2 congruent ones. Then, for the larger, dark blue triangle similar to them, I did another " copy and paste", changed the color to darker blue, and, with the grid showing on the drawing canvas, I stretched the small triangle, both horizontally and vertically, until the base and height were twice as big as before.

To create the green rectangles I "drew" the big one first, then I copied it and, using the grid, I shrunk or reduced the copy to half it's original size. In both cases, the ratio of the larger side to the smaller side is 2 to 1 and the ratio of the smaller side to the larger side is 1 to 2.

 Geometric figures are similar if a) corresponding angles are equal, b) corresponding sides are proportional.

In the pentagons, the corresponding angles are equal, so, the sides are proportional.

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Let's look at the 2 rectangles again:

The perimeter of the large green rectangle is 20 units, the smaller rectangle's perimeter = 10 units.
The
larger perimeter is twice the smaller. The smaller is half of the larger.

In similar figures, the ratio of perimeters is equal to the ratio of sides.

The area of the large green rectangle is 8 × 2 or 16 square units. The area of the small green rectangle is 4 × 1 or 4 square units.

The ratio of the areas is equal to the square of the ratio of the sides.

The ratio of the sides is called the proportionality factor.

We generally use c or k to represent it because
it can be called the proportionality CONSTANT.

For our rectangles, c = 2 for big side to little side and c = ½ for little side to big side.

Example:

1) Use the information in the diagram to answer the questions.

 a) Find angles C, Q, P and R? a) C = R = 64°, Q = B = 30°, P = A = 86°. b) Find the lengths of PQ and PR. b) , PQ = 2/3(5.7) = 3.8 in PR = 2.2 in c) What is the ratio of BC to QR? c) BC to QR = 3 to 2 d) What is the ratio of PQ to AB? d) PQ to AB = 2 to 3 e) What is the perimeter of ABC? of PQR? e) P for ABC = 15 inches, P for PQR = 10 inches f) Find the "big to small" perimeters ratio. f) 15 to 10 = 3 to 2. g) Find the area of each triangle. g) Area ABC = ½(6 × 3) = 9 in², PQR = 4 in² h) Find the "small to big" ratio of the areas? h) small to big ratio = 4 to 9 or (2²) to (3²).

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Triangle Similarity Theorems

(AA): If 2 angles of one triangle are equal to 2 angles of another, the triangles are similar.
(the third angles are also equal because the sum of 3 angles = 180°)

(SSS): If the sides of one triangle are respectively proportional to the sides of another,
the triangles are similar. (respectively means the sides are corresponding -- same position.)

(SAS): If 2 sides of one triangle are proportional to 2 sides of another, and the
angles contained between them are equal, the triangles are similar.

Be careful with this one -- the angle must be between the proportional sides.

Line Drawn Parallel to a Side

Any line drawn parallel to a side of a triangle creates similar triangles.

Examples:Similar triangles and the proportions of the corresponding sides.

In the examples above, it's not always immediately obvious which sides are proportional to which in the two similar figures. If we mark the corresponding angles, we can see which sides are adjacent to or "contained by" corresponding angles. Let's look at the 3rd example above.

Once we prove the figures are similar, we mark the corresponding angles with letters or numbers and then it's easy to see the corresponding sides.

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Now get a pencil, an eraser and a note book, copy the questions,
do the practice exercise(s),then check your work with the solutions.
If you get stuck, review the examples in the lesson, then try again.

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Practice Exercise

1) On grid (graph) paper, draw triangle TNE with angle T = 90°, TN = 12 squares, TE = 9.
Beside it, draw triangle CFA with angle C = 90° and CF = 8 squares, CA = 6.
a) Use a ruler to measure the lengths of the corresponding sides of the 2 triangles.
b) Use a protractor to measure the acute angles in both triangles.
c) What is the ratio of the sides of triangles TNE and CFA?
d) Find the area of triangle CFA.
e) What is the ratio of the areas of triangle TNE to triangle CFA? Explain.

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3) Use the diagram to find the height of the tree.

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Solutions

1)
a) Measuring in inches on ¼ inch grid paper, TN = 3 inches, TE = 2¼ inches, EN = 3¾ inches.
CF = 2 inches, CA = 1½ inches, AF = 2½ inches.

b) angle E = 53° = angle A, angle N = 37° = angle F. These angles are complementary.

c) The ratio of the sides is 3 to 2.

d) Area = ½ b × h = ½ (1½)(2) = 1½ inches².

e) The area of triangle TNE is 2¼ times the area of triangle CFA.
ratio of the sides = (3 : 2), so ratio of the areas = 9 : 4.

2)

 a) EF = 2, FG = 6, EI = 3, IH = 9 b) EF : EG = 2 : 8 = 1 : 4 c) area ratio = 1 : 16

3) Use the diagram to find the height of the tree.

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