a triangle abc has side lengths of a= 14 cm, b=20cm, c=7.5cm. find the size of the internal angles and the area of the triangle

To solve this problem, we will apply **the law of cosines** to find the internal angles of the triangle and then use **Heron's formula** to calculate the area of the triangle.

### Step 1: Calculate the angles using the Law of Cosines

The Law of Cosines states that for a triangle with sides \(a\), \(b\), and \(c\):

\[
\cos A = \frac{b^2 + c^2 - a^2}{2bc}
\]
\[
\cos B = \frac{a^2 + c^2 - b^2}{2ac}
\]
\[
\cos C = \frac{a^2 + b^2 - c^2}{2ab}
\]

Where:
- \(A\), \(B\), and \(C\) are the angles opposite sides \(a\), \(b\), and \(c\), respectively.

### Step 2: Calculate the area using Heron's formula

Heron's formula gives the area of a triangle when the lengths of all three sides are known:

\[
A = \sqrt{s(s-a)(s-b)(s-c)}
\]

Where \(s\) is the semi-perimeter of the triangle:

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

Let's calculate the angles and the area now.The internal angles of the triangle are approximately:

- \( A = 29.83^\circ \)
- \( B = 134.71^\circ \)
- \( C = 15.46^\circ \)

The area of the triangle is approximately **37.31 cm²**.

Would you like more details or have any questions? 

Here are some related questions to deepen understanding:
1. How does the Law of Cosines apply in triangles with different side lengths?
2. Can you explain why Heron's formula works for all triangles?
3. What happens if the triangle is a right triangle? How do these formulas change?
4. Could we calculate the area using the sine formula instead of Heron's formula?
5. How does the size of the angles influence the shape of the triangle?

**Tip:** Always verify that the sum of the internal angles in a triangle adds up to 180 degrees to confirm your calculations are correct.

A triangle ABC has side lengths of a = 14 cm, b = 20 cm, c = 7.5 cm. Find the size of the internal angles and the area of the triangle.

Learn how to find the internal angles and area of a triangle with side lengths of 14 cm, 20 cm, and 7.5 cm using the Law of Cosines and Heron's Formula. This problem is ideal for students in Grades 10-12 and covers essential geometric and trigonometric concepts.

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