To make an accurate drawing of the triangle and find the length $d$: Instructions for drawing the triangle: 1. Draw a line segment of length $8.9$ cm. This will be the base of your triangle. 2. At one end of the $8.9$ cm line segment, place the center of a protractor. Measure and mark an angle of $78^\circ$. 3. From the vertex where you measured the $78^\circ$ angle, draw a line segment of length $7.5$ cm along the $78^\circ$ mark. 4. Connect the end of the $8.9$ cm line segment to the end of the $7.5$ cm line segment. This third side is the length $d$. 5. Measure the length of side $d$ with a ruler and round it to 1 decimal place. Calculation of length $d$ using the Law of Cosines: Since we are given two sides and the included angle (SAS), we can use the Law of Cosines: $$ d^2 = a^2 + b^2 - 2ab \cos(\theta) $$ Here, $a = 7.5$ cm, $b = 8.9$ cm, and $\theta = 78^\circ$. Step 1: Substitute the values into the formula. $$ d^2 = (7.5)^2 + (8.9)^2 - 2(7.5)(8.9) \cos(78^\circ) $$ Step 2: Calculate the squares of the sides. $$ (7.5)^2 = 56.25 $$ $$ (8.9)^2 = 79.21 $$ Step 3: Calculate the product $2ab$. $$ 2(7.5)(8.9) = 15 \times 8.9 = 133.5 $$ Step 4: Find the value of $\cos(78^\circ)$. $$ \cos(78^\circ) \approx 0.20791169 $$ Step 5: Substitute these values back into the equation for $d^2$. $$ d^2 = 56.25 + 79.21 - 133.5 \times 0.20791169 $$ $$ d^2 = 135.46 - 27.750259 $$ $$ d^2 = 107.709741 $$ Step 6: Take the square root to find $d$. $$ d = \sqrt{107.709741} $$ $$ d \approx 10.37833 $$ Step 7: Round the length $d$ to 1 decimal place. $$ d \approx 10.4 \text{ cm} $$ The measured length $d$ in your drawing should be approximately $10.4$ cm. $$ d = \boxed{\text{10.4 cm}} $$