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Identify the given sides of the right-angled triangle ABC.

Mathematics

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2 parts
a

24 cm24\,\text{cm}

b

84 cm284\,\text{cm}^2

Step-by-step solution

8 steps

Part a· 5 steps

Determine the length of side AC.

  1. Identify the given sides of the right-angled triangle ABC.

    The hypotenuse is BC = 25 cm.
    One leg is AB = 7 cm.
    The other leg is AC.

  2. Apply the Pythagorean theorem.

    For a right-angled triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides:

    AC2+AB2=BC2AC^2 + AB^2 = BC^2

  3. Substitute the known values into the equation.

    AC2+(7 cm)2=(25 cm)2AC^2 + (7\,\text{cm})^2 = (25\,\text{cm})^2 AC2+49 cm2=625 cm2AC^2 + 49\,\text{cm}^2 = 625\,\text{cm}^2

  4. Solve for AC2AC^2.

    AC2=625 cm2−49 cm2AC^2 = 625\,\text{cm}^2 - 49\,\text{cm}^2 AC2=576 cm2AC^2 = 576\,\text{cm}^2

  5. Calculate AC by taking the square root.

    AC=576 cm2AC = \sqrt{576\,\text{cm}^2} AC=24 cmAC = 24\,\text{cm}

    The length of side AC is 24 cm\boxed{24\,\text{cm}}.

Part b

84 cm284\,\text{cm}^2

Determine the area of the design.

  1. Identify the base and height of the triangle.

    For a right-angled triangle, the legs can be considered the base and height.
    Base (AB) = 7 cm
    Height (AC) = 24 cm (from part a)

  2. Use the formula for the area of a triangle.

    Area=12×base×height\text{Area} = \frac{1}{2} \times \text{base} \times \text{height}

  3. Substitute the values and calculate the area.

    Area=12×7 cm×24 cm\text{Area} = \frac{1}{2} \times 7\,\text{cm} \times 24\,\text{cm} Area=12×168 cm2\text{Area} = \frac{1}{2} \times 168\,\text{cm}^2 Area=84 cm2\text{Area} = 84\,\text{cm}^2

    The area of the design is 84 cm2\boxed{84\,\text{cm}^2}.

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