To construct triangle ABC based on the given information:

• $AB = 6$ cm,
• $\angle BAC = 96^\circ$,
• $\angle ABC = 35^\circ$.

Here’s the solution method:

Steps for construction:

1. Draw the base $AB$: Draw a line segment of 6 cm.
2. Construct angle $\angle BAC = 96^\circ$: Use a protractor to measure an angle of 96° at vertex $A$ (with vertex $A$ being at the intersection of the two sides of the angle).
3. Construct angle $\angle ABC = 35^\circ$: Use a protractor to measure an angle of 35° at vertex $B$.
4. Locate point C: Extend the lines from A and B based on the angles, and the point where they intersect will be the third vertex $C$.
5. Measure side BC: Use a ruler to measure the length of side $BC$.

Solving the triangle:

After constructing the triangle, we can apply the law of sines to calculate the length of $BC$.

Law of Sines:

$\frac{BC}{\sin \angle BAC} = \frac{AB}{\sin \angle ACB}$ First, calculate $\angle ACB$. Since the sum of angles in a triangle is 180°, we have: $\angle ACB = 180^\circ - \angle BAC - \angle ABC = 180^\circ - 96^\circ - 35^\circ = 49^\circ$ Now apply the law of sines: $\frac{BC}{\sin 96^\circ} = \frac{6}{\sin 49^\circ}$ Solving for $BC$: $BC = \frac{6 \cdot \sin 96^\circ}{\sin 49^\circ}$ We can compute this value to find the length of side $BC$.

Let me calculate this for you! math

Given values

AB = 6 # length of AB in cm angle_BAC = 96 # angle BAC in degrees angle_ABC = 35 # angle ABC in degrees

Calculate angle ACB

angle_ACB = 180 - angle_BAC - angle_ABC

Convert angles to radians for the sine function

angle_BAC_rad = math.radians(angle_BAC) angle_ACB_rad = math.radians(angle_ACB) angle_ABC_rad = math.radians(angle_ABC)

Law of sines to find BC

BC = (AB * math.sin(angle_ACB_rad)) / math.sin(angle_BAC_rad)

Round BC to 1 decimal place

BC_rounded = round(BC, 1) BC_roundedThe length of side $BC$ is approximately 4.6 cm (rounded to 1 decimal place).

Would you like further details or have any questions about this solution? Here are some related questions to explore:

1. How do you use the law of sines in different types of triangles?
2. What are the conditions for applying the law of sines?
3. Can you solve the triangle using the law of cosines instead of the law of sines?
4. How do you measure angles accurately using a protractor for construction?
5. What other triangle properties can help solve unknown sides and angles?

Tip: Always double-check your angle measurements during construction to ensure accuracy.