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Given the sides of the triangle are a = 13 cm, b = 14 cm, and c = 15 cm.

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4 parts
a

21 cm\text{21 cm}

b

8 cm\text{8 cm}

7 cm\text{7 cm}

6 cm\text{6 cm}

c

84.00 cm2\text{84.00 cm}^2

d

336 cm2\text{336 cm}^2

Step-by-step solution

11 steps

here's the solution for Question 4:

Given the sides of the triangle are a=13 cma = 13 \text{ cm}, b=14 cmb = 14 \text{ cm}, and c=15 cmc = 15 \text{ cm}.

Part a· 2 steps

Calculate the semi-perimeter ss.

  1. Sum the lengths of the sides.

    a+b+c=13 cm+14 cm+15 cm=42 cma + b + c = 13 \text{ cm} + 14 \text{ cm} + 15 \text{ cm} = 42 \text{ cm}

  2. Divide the sum by 2 to find the semi-perimeter.

    s=a+b+c2=42 cm2=21 cms = \frac{a + b + c}{2} = \frac{42 \text{ cm}}{2} = 21 \text{ cm}

    The semi-perimeter ss is 21 cm\boxed{\text{21 cm}}.

Part b

8 cm\text{8 cm}

Calculate (s−a)(s - a), (s−b)(s - b), and (s−c)(s - c).

  1. Calculate (s−a)(s - a).

    s−a=21 cm−13 cm=8 cms - a = 21 \text{ cm} - 13 \text{ cm} = 8 \text{ cm}

  2. Calculate (s−b)(s - b).

    s−b=21 cm−14 cm=7 cms - b = 21 \text{ cm} - 14 \text{ cm} = 7 \text{ cm}

  3. Calculate (s−c)(s - c).

    s−c=21 cm−15 cm=6 cms - c = 21 \text{ cm} - 15 \text{ cm} = 6 \text{ cm}

    The values are (s−a)=8 cm(s - a) = \boxed{\text{8 cm}}, (s−b)=7 cm(s - b) = \boxed{\text{7 cm}}, and (s−c)=6 cm(s - c) = \boxed{\text{6 cm}}.

Part c

84.00 cm2\text{84.00 cm}^2

Apply Heron's Formula to find the area.

  1. Substitute the values into Heron's Formula.

    Area=s(s−a)(s−b)(s−c)\text{Area} = \sqrt{s(s-a)(s-b)(s-c)} Area=21 cm×8 cm×7 cm×6 cm\text{Area} = \sqrt{21 \text{ cm} \times 8 \text{ cm} \times 7 \text{ cm} \times 6 \text{ cm}}

  2. Multiply the terms under the square root.

    Area=7056 cm4\text{Area} = \sqrt{7056 \text{ cm}^4}

  3. Calculate the square root.

    Area=84 cm2\text{Area} = 84 \text{ cm}^2

    The area of the triangle is 84.00 cm2\boxed{\text{84.00 cm}^2} (correct to 2 decimal places).

Part d

336 cm2\text{336 cm}^2

Find the new area if all three sides are doubled.

  1. Determine the linear scale factor.

    The sides are doubled, so the linear scale factor k=2k = 2.

  2. Determine the area scale factor.

    The area scale factor is k2k^2.

    Area scale factor=(2)2=4\text{Area scale factor} = (2)^2 = 4

  3. Multiply the original area by the area scale factor to find the new area.

    New Area=Original Area×Area scale factor\text{New Area} = \text{Original Area} \times \text{Area scale factor} New Area=84 cm2×4\text{New Area} = 84 \text{ cm}^2 \times 4 New Area=336 cm2\text{New Area} = 336 \text{ cm}^2

    The new area is 336 cm2\boxed{\text{336 cm}^2}.

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