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A triangle ABC has AB = 8 cm, AC = 10 cm, and angle A = 30 degrees. Find the area of the triangle.

Mathematics

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Answers

3 parts
a

13 cm\text{13 cm}

b

30 cm2\text{30 cm}^2

c

30 cm\text{30 cm}

Step-by-step solution

9 steps

Based on the image, it appears to be a right-angled triangle ABC, with the right angle at B. The given side lengths are AB = 12 cm and BC = 5 cm. You need to calculate the length of AC, the area, and the perimeter of the triangle.

Part a· 3 steps

Calculate AC

  1. Use the Pythagorean theorem, which states that for a right-angled triangle, the square of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides (a2+b2=c2a^2 + b^2 = c^2).

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

  2. Substitute the given values into the formula.

    AC2=(12 cm)2+(5 cm)2AC^2 = (12 \text{ cm})^2 + (5 \text{ cm})^2 AC2=144 cm2+25 cm2AC^2 = 144 \text{ cm}^2 + 25 \text{ cm}^2

  3. Add the values and find the square root to get the length of AC.

    AC2=169 cm2AC^2 = 169 \text{ cm}^2 AC=169 cm2AC = \sqrt{169 \text{ cm}^2} AC=13 cmAC = \boxed{\text{13 cm}}

Part b

30 cm2\text{30 cm}^2

Calculate the area of triangle ABC

  1. The area of a right-angled triangle is given by the formula 12×base×height\frac{1}{2} \times \text{base} \times \text{height}.

    In this case, AB and BC are the base and height.

    Area=12×AB×BC\text{Area} = \frac{1}{2} \times AB \times BC

  2. Substitute the known values into the formula.

    Area=12×12 cm×5 cm\text{Area} = \frac{1}{2} \times 12 \text{ cm} \times 5 \text{ cm}

  3. Perform the multiplication to find the area.

    Area=12×60 cm2\text{Area} = \frac{1}{2} \times 60 \text{ cm}^2 Area=30 cm2\text{Area} = \boxed{\text{30 cm}^2}

Part c

30 cm\text{30 cm}

Calculate the perimeter of triangle ABC

  1. The perimeter of a triangle is the sum of the lengths of all its sides.

    Perimeter=AB+BC+AC\text{Perimeter} = AB + BC + AC

  2. Substitute the lengths of the sides (including AC calculated in part a) into the formula.

    Perimeter=12 cm+5 cm+13 cm\text{Perimeter} = 12 \text{ cm} + 5 \text{ cm} + 13 \text{ cm}

  3. Add the lengths to find the total perimeter.

    Perimeter=30 cm\text{Perimeter} = \boxed{\text{30 cm}}
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