Heron's formula class 9 worksheet

Mathematics
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heron’s formula class 9 worksheet

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Heron’s Formula Class 9 Worksheet

Answer:
Heron’s formula is a critical topic in mathematics for class 9 students, assisting in solving problems related to finding the area of a triangle when the lengths of all three sides are known. Let’s delve into detailed explanations and steps for understanding and applying Heron’s formula. Below, you’ll find a comprehensive guide along with a sample worksheet tailored for class 9 students.

Heron’s Formula

Definition:
Heron’s formula states that the area of a triangle whose sides have lengths a, b, and c, can be calculated using the formula:

Area = \sqrt{s(s-a)(s-b)(s-c)}

where s is the semi-perimeter of the triangle, calculated as:

s = \frac{a+b+c}{2}

Solution By Steps:

  1. Calculate the Semi-Perimeter, ( s ):

    • Sum the lengths of all three sides.

    • Divide the sum by 2 to get the semi-perimeter.

      s = \frac{a+b+c}{2}

  2. Apply Heron’s Formula:

    • Substitute the semi-perimeter and the side lengths into Heron’s formula.

      Area = \sqrt{s(s-a)(s-b)(s-c)}

  3. Solve the Expression:

    • Simplify inside the square root.
    • Compute the square root to find the area.

Sample Worksheet Problems

Problem 1:

Find the area of a triangle with sides 7 cm, 8 cm, and 9 cm.

Solution:

  1. Calculate the Semi-Perimeter ( s ):

    s = \frac{a+b+c}{2} = \frac{7+8+9}{2} = 12

  2. Apply Heron’s Formula:

    Area = \sqrt{12(12-7)(12-8)(12-9)} = \sqrt{12 \cdot 5 \cdot 4 \cdot 3}

  3. Solve the Expression:

    Area = \sqrt{12 \cdot 5 \cdot 4 \cdot 3} = \sqrt{720} = \sqrt{36 \cdot 20} = 6\sqrt{20} \approx 26.83 \text{ cm}^2

Problem 2:

Find the area of a triangle with sides 13 cm, 14 cm, and 15 cm.

Solution:

  1. Calculate the Semi-Perimeter ( s ):

    s = \frac{13+14+15}{2} = 21

  2. Apply Heron’s Formula:

    Area = \sqrt{21(21-13)(21-14)(21-15)} = \sqrt{21 \cdot 8 \cdot 7 \cdot 6}

  3. Solve the Expression:

    Area = \sqrt{7056} = 84 \text{ cm}^2

Problem 3:

Calculate the area of a triangle with sides 5 cm, 6 cm, and 7 cm.

Solution:

  1. Calculate the Semi-Perimeter ( s ):

    s = \frac{5+6+7}{2} = 9

  2. Apply Heron’s Formula:

    Area = \sqrt{9(9-5)(9-6)(9-7)} = \sqrt{9 \cdot 4 \cdot 3 \cdot 2}

  3. Solve the Expression:

    Area = \sqrt{216} = 6\sqrt{6} \approx 14.7 \text{ cm}^2

Practice Problems

  1. Find the area of a triangle with sides 10 cm, 10 cm, and 12 cm.
  2. Find the area of a triangle with sides 11 cm, 13 cm, and 16 cm.
  3. Calculate the area of a triangle with sides 8 cm, 15 cm, and 17 cm.

Conclusion

Heron’s formula is an essential tool for calculating the area of triangles when the lengths of the sides are known. By following the steps outlined above, students can confidently solve problems using Heron’s formula. This worksheet provides practice problems to help reinforce the concept.

Final Note:
If you have any more questions or need further assistance with Heron’s formula, feel free to ask. Happy studying!