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Polygons

Polygons are all around us. A stop sign is in the shape of an octagon, an eight-sided polygon. The honeycomb of a beehive consists of hexagonal cells. The top of a desk is a rectangle.

In this section, you will learn about similar polygons.

Similarity of Polygons

Discussion : Similar Triangles

Fill in the table using the diagram and then answer the questions that follow.

Table 1
AB DE AB DE =...cm...cm=......cm...cm=... A^A^=...∘∘ D^D^...∘∘
BC EF BC EF =...cm...cm=......cm...cm=... B^B^=...∘∘ E^E^=...∘∘
AC DF AC DF =...cm...cm=......cm...cm=... C^C^...∘∘ F^F^=...∘∘

Figure 1
Figure 1 (MG10C14_010.png)

  1. What can you say about the numbers you calculated for: AB DE AB DE , BC EF BC EF , AC DF AC DF ?
  2. What can you say about A^A^ and D^D^?
  3. What can you say about B^B^ and E^E^?
  4. What can you say about C^C^ and F^F^?

If two polygons are similar, one is an enlargement of the other. This means that the two polygons will have the same angles and their sides will be in the same proportion.

We use the symbol ≡≡ to mean is similar to.

Definition 1: Similar Polygons

Two polygons are similar if:

  1. their corresponding angles are equal, and
  2. the ratios of corresponding sides are equal.

Exercise 1: Similarity of Polygons

Show that the following two polygons are similar.

Figure 2
Figure 2 (MG10C14_011.png)

Solution
  1. Step 1. Determine what is required :

    We are required to show that the pair of polygons is similar. We can do this by showing that the ratio of corresponding sides is equal and by showing that corresponding angles are equal.

  2. Step 2. Corresponding angles :

    We are given the angles. So, we can show that corresponding angles are equal.

  3. Step 3. Show that corresponding angles are equal :

    All angles are given to be 90∘∘ and

    A ^ = E ^ B ^ = F ^ C ^ = G ^ D ^ = H ^ A ^ = E ^ B ^ = F ^ C ^ = G ^ D ^ = H ^
    (1)
  4. Step 4. Show that corresponding sides have equal ratios :

    We first need to see which sides correspond. The rectangles have two equal long sides and two equal short sides. We need to compare the ratio of the long side lengths of the two different rectangles as well as the ratio of the short side lenghts.

    Long sides, large rectangle values over small rectangle values:

    Ratio = 2 L L = 2 Ratio = 2 L L = 2
    (2)

    Short sides, large rectangle values over small rectangle values:

    Ratio = L 1 2 L = 1 1 2 = 2 Ratio = L 1 2 L = 1 1 2 = 2
    (3)

    The ratios of the corresponding sides are equal, 2 in this case.

  5. Step 5. Final answer :

    Since corresponding angles are equal and the ratios of the corresponding sides are equal the polygons ABCD and EFGH are similar.

Tip:

All squares are similar.

Exercise 2: Similarity of Polygons

If two pentagons ABCDE and GHJKL are similar, determine the lengths of the sides and angles labelled with letters:

Figure 3
Figure 3 (MG10C14_012.png)

Solution
  1. Step 1. Determine what is given :

    We are given that ABCDE and GHJKL are similar. This means that:

    AB GH = BC HJ = CD JK = DE KL = EA LG AB GH = BC HJ = CD JK = DE KL = EA LG
    (4)

    and

    A ^ = G ^ B ^ = H ^ C ^ = J ^ D ^ = K ^ E ^ = L ^ A ^ = G ^ B ^ = H ^ C ^ = J ^ D ^ = K ^ E ^ = L ^
    (5)
  2. Step 2. Determine what is required :

    We are required to determine the

    1. lengths: aa, bb, cc and dd, and
    2. angles: ee, ff and gg.
  3. Step 3. Decide how to approach the problem :

    The corresponding angles are equal, so no calculation is needed. We are given one pair of sides DCDC and KJKJ that correspond. DCKJ=4,53=1,5DCKJ=4,53=1,5 so we know that all sides of KJHGLKJHGL are 1,5 times smaller than ABCDEABCDE.

  4. Step 4. Calculate lengths :
    a 2 = 1 , 5 ∴ a = 2 × 1 , 5 = 3 b 1 , 5 = 1 , 5 ∴ b = 1 , 5 × 1 , 5 = 2 , 25 6 c = 1 , 5 ∴ c = 6 ÷ 1 , 5 = 4 d = 3 1 , 5 ∴ d = 2 a 2 = 1 , 5 ∴ a = 2 × 1 , 5 = 3 b 1 , 5 = 1 , 5 ∴ b = 1 , 5 × 1 , 5 = 2 , 25 6 c = 1 , 5 ∴ c = 6 ÷ 1 , 5 = 4 d = 3 1 , 5 ∴ d = 2
    (6)
  5. Step 5. Calculate angles :
    e = 92 ∘ ( corresponds to H ) f = 120 ∘ ( corresponds to D ) g = 40 ∘ ( corresponds to E ) e = 92 ∘ ( corresponds to H ) f = 120 ∘ ( corresponds to D ) g = 40 ∘ ( corresponds to E )
    (7)
  6. Step 6. Write the final answer :
    a = 3 b = 2 , 25 c = 4 d = 2 e = 92 ∘ f = 120 ∘ g = 40 ∘ a = 3 b = 2 , 25 c = 4 d = 2 e = 92 ∘ f = 120 ∘ g = 40 ∘
    (8)

Similarity of Equilateral Triangles

Working in pairs, show that all equilateral triangles are similar.

Polygons-mixed

  1. Find the values of the unknowns in each case. Give reasons.
    Figure 4
    Figure 4 (mg10c14_2.png)
    Click here for the solution
  2. Find the angles and lengths marked with letters in the following figures:
    Figure 5
    Figure 5 (MG10C14_014.png)
    Click here for the solution

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