Problem

Compare the triangle areas when the \(\text{same}~\text{two}~\text{sides}\), \(10\) and \(12\), are fixed and the included angles are \(60^{\circ}\) and \(90^{\circ}\).

Two triangles with the same side lengths 10 and 12 and included angles 60 and 90 degrees, each with a perpendicular altitude.

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The triangle with a 90° included angle has the larger area because the area formula becomes 1/2abC, so 90 makes a bigger area than 60.

The two triangles have the same area because both use the same side lengths 10 and 12, so changing only the included angle does not affect area.

The triangle with a 90° included angle has the larger area because with sides 10 and 12 fixed, area is proportional to sin(C), and sin 90° is greater than sin 60°.

The triangle with a 60° included angle has the larger area because acute triangles keep the altitude inside, while the 90° case loses height.

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Compare the triangle areas when the \(\text{same}~\text{two}~\text{sides}\), \(10\) and \(12\), are fixed and the included angles are \(60^{\circ}\) and \(90^{\circ}\).

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