Construct a triangle ABC with AB = 5 cm, BC = 6 cm and angle B = 60 degrees and find its area.

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Step-by-Step Solution

Step 1: Construct the base and angle

First, we draw the base of the triangle, segment $BC$, with a length of 6 cm. Then, at point $B$, we construct an angle of $60^\circ$ using a protractor or compass, extending a ray $BX.

Step 2: Mark point A

Next, we measure 5 cm along the ray $BX$ from point $B$ and mark point $A$. Finally, we connect points $A$ and $C$ to complete the triangle $ABC.

Step 3: Calculate the area of the triangle

The area of a triangle can be calculated using the formula $\frac{1}{2}ab\sin C$, where $a$ and $b$ are two sides and $C$ is the included angle. Here, we use sides $AB$ and $BC$ and the included angle $B.

Step 4: Substitute values and simplify

We substitute the given values: $AB = 5$ cm, $BC = 6$ cm, and $\sin(60^\circ) = \frac{\sqrt{3}}{2}$ into the area formula. After performing the multiplication, we get the area as $7.5\sqrt{3}$ square centimeters.