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Given that the triangles are similar, the ratio of corresponding sides is equal.

Mathematics

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3 parts
21

9 cm\text{9 cm}

22

15 cm\text{15 cm}

23

32 cm\text{32 cm}

Step-by-step solution

9 steps

Here are the solutions to questions 21, 22, and 23.

Part 21· 3 steps

  1. Identify corresponding sides.

    Given that the triangles are similar, the ratio of corresponding sides is equal.
    Side ABAB corresponds to side DEDE.
    Side BCBC corresponds to side EFEF.

  2. Set up the proportion and substitute known values.

    ABDE=BCEF\frac{AB}{DE} = \frac{BC}{EF} 6 cmDE=8 cm12 cm\frac{6\, \text{cm}}{DE} = \frac{8\, \text{cm}}{12\, \text{cm}}

  3. Solve for DEDE.

    6×12=8×DE6 \times 12 = 8 \times DE 72=8×DE72 = 8 \times DE DE=728DE = \frac{72}{8} DE=9 cmDE = 9\, \text{cm}

    The length of side DEDE is 9 cm\boxed{\text{9 cm}}.

Part 22

15 cm\text{15 cm}

  1. Understand the given ratio.

    The ratio of corresponding sides of the smaller triangle to the larger triangle is 3:53:5.
    Let SsS_s be the side of the smaller triangle and SlS_l be the corresponding side of the larger triangle.

    SsSl=35\frac{S_s}{S_l} = \frac{3}{5}

  2. Substitute the given side of the smaller triangle.

    Given Ss=9 cmS_s = 9\, \text{cm}.

    9 cmSl=35\frac{9\, \text{cm}}{S_l} = \frac{3}{5}

  3. Solve for SlS_l.

    9×5=3×Sl9 \times 5 = 3 \times S_l 45=3×Sl45 = 3 \times S_l Sl=453S_l = \frac{45}{3} Sl=15 cmS_l = 15\, \text{cm}

    The corresponding side of the larger triangle is 15 cm\boxed{\text{15 cm}}.

Part 23

32 cm\text{32 cm}

  1. Identify corresponding sides and the given ratio.

    The triangles △XYZ\triangle XYZ and △DEF\triangle DEF are similar.
    Side XYXY corresponds to side DEDE.
    The ratio of corresponding sides is 3:123:12. Assuming the first number corresponds to the smaller triangle and the second to the larger.

    XYDE=312\frac{XY}{DE} = \frac{3}{12}

  2. Substitute the given length of XYXY.

    Given XY=8 cmXY = 8\, \text{cm}.

    8 cmDE=312\frac{8\, \text{cm}}{DE} = \frac{3}{12}

  3. Solve for DEDE.

    8×12=3×DE8 \times 12 = 3 \times DE 96=3×DE96 = 3 \times DE DE=963DE = \frac{96}{3} DE=32 cmDE = 32\, \text{cm}

    The length of side DEDE is 32 cm\boxed{\text{32 cm}}.

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