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Problem 47: Calculate the measure of the fourth side of a quadrilateral given the perimeter (2p) and the measures of the other three sides (a, b, c).

Mathematics

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5 parts
a

7 cm7 \text{ cm}

b

40 cm40 \text{ cm}

c

65 cm65 \text{ cm}

d

80 cm80 \text{ cm}

e

15 cm15 \text{ cm}

Step-by-step solution

10 steps

The image contains several math problems. I will solve , which asks to calculate the measure of the fourth side of the following quadrilaterals, given the perimeter (2p2p) and the measures of the other sides (a,b,ca, b, c).

The perimeter of a quadrilateral is the sum of the lengths of its four sides. Let the four sides be s1,s2,s3,s4s_1, s_2, s_3, s_4. The perimeter 2p2p is given by:

2p=s1+s2+s3+s42p = s_1 + s_2 + s_3 + s_4

To find the fourth side, we can rearrange the formula:

s4=2p−(s1+s2+s3)s_4 = 2p - (s_1 + s_2 + s_3)

Part a· 2 steps

Given 2p=352p = 35 cm, a=5a = 5 cm, b=8b = 8 cm, c=15c = 15 cm.

  1. Sum the known sides.

    5 cm+8 cm+15 cm=28 cm5 \text{ cm} + 8 \text{ cm} + 15 \text{ cm} = 28 \text{ cm}

  2. Subtract the sum from the perimeter.

    s4=35 cm−28 cm=7 cms_4 = 35 \text{ cm} - 28 \text{ cm} = 7 \text{ cm}

    The fourth side is 7 cm\boxed{7 \text{ cm}}.

Part b

40 cm40 \text{ cm}

Given 2p=852p = 85 cm, a=14a = 14 cm, b=15b = 15 cm, c=16c = 16 cm.

  1. Sum the known sides.

    14 cm+15 cm+16 cm=45 cm14 \text{ cm} + 15 \text{ cm} + 16 \text{ cm} = 45 \text{ cm}

  2. Subtract the sum from the perimeter.

    s4=85 cm−45 cm=40 cms_4 = 85 \text{ cm} - 45 \text{ cm} = 40 \text{ cm}

    The fourth side is 40 cm\boxed{40 \text{ cm}}.

Part c

65 cm65 \text{ cm}

Given 2p=2002p = 200 cm, a=35a = 35 cm, b=45b = 45 cm, c=55c = 55 cm.

  1. Sum the known sides.

    35 cm+45 cm+55 cm=135 cm35 \text{ cm} + 45 \text{ cm} + 55 \text{ cm} = 135 \text{ cm}

  2. Subtract the sum from the perimeter.

    s4=200 cm−135 cm=65 cms_4 = 200 \text{ cm} - 135 \text{ cm} = 65 \text{ cm}

    The fourth side is 65 cm\boxed{65 \text{ cm}}.

Part d

80 cm80 \text{ cm}

Given 2p=2102p = 210 cm, a=30a = 30 cm, b=40b = 40 cm, c=60c = 60 cm.

  1. Sum the known sides.

    30 cm+40 cm+60 cm=130 cm30 \text{ cm} + 40 \text{ cm} + 60 \text{ cm} = 130 \text{ cm}

  2. Subtract the sum from the perimeter.

    s4=210 cm−130 cm=80 cms_4 = 210 \text{ cm} - 130 \text{ cm} = 80 \text{ cm}

    The fourth side is 80 cm\boxed{80 \text{ cm}}.

Part e

15 cm15 \text{ cm}

Given 2p=2122p = 212 cm, a=27a = 27 cm, b=71b = 71 cm, c=99c = 99 cm.

  1. Sum the known sides.

    27 cm+71 cm+99 cm=197 cm27 \text{ cm} + 71 \text{ cm} + 99 \text{ cm} = 197 \text{ cm}

  2. Subtract the sum from the perimeter.

    s4=212 cm−197 cm=15 cms_4 = 212 \text{ cm} - 197 \text{ cm} = 15 \text{ cm}

    The fourth side is 15 cm\boxed{15 \text{ cm}}.

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