Find the area of each triangle using the formula 𝓐 = ½ bh, and then verify that the formula 𝓐 = ½ ab sin C gives the same result.
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Verified step by step guidance

1. Identify the base (b) and height (h) of the triangle from the given image.
2. Use the formula \( \mathcal{A} = \frac{1}{2} b h \) to calculate the area of the triangle.
3. Identify the sides (a and b) and the included angle (C) from the given image.
4. Use the formula \( \mathcal{A} = \frac{1}{2} ab \sin C \) to calculate the area of the triangle.
5. Compare the results from both formulas to verify they give the same area.

Key Concepts

Area of a Triangle

The area of a triangle can be calculated using the formula A = ½ bh, where 'b' is the base and 'h' is the height. This formula is derived from the fact that a triangle is essentially half of a rectangle formed by the base and height. Understanding this formula is crucial for calculating the area of triangles in various contexts.

Trigonometric Functions

Trigonometric functions, such as sine, cosine, and tangent, relate the angles of a triangle to the lengths of its sides. The formula A = ½ ab sin C uses the sine of angle C to find the area of a triangle when two sides and the included angle are known. This concept is essential for solving problems involving non-right triangles.

Verification of Area Formulas

Verifying that different formulas yield the same area involves substituting known values and ensuring consistency between the results. In this case, comparing A = ½ bh with A = ½ ab sin C demonstrates the relationship between the height and the sine of the angle in a triangle. This verification reinforces the understanding of how different geometric properties are interconnected.