Make an accurate drawing of the triangle below. Measure length d in your drawing to 1 d.p.

Question

ScanSolve AI Tutor · Accepted Answer

To make an accurate drawing of the triangle and find the length $d$ :

Instructions for drawing the triangle:

Draw a line segment of length 8.9 cm. This will be the base of your triangle.
At one end of the 8.9 cm line segment, place the center of a protractor. Measure and mark an angle of $78^\circ$ .
From the vertex where you measured the $78^\circ$ angle, draw a line segment of length 7.5 cm along the $78^\circ$ mark.
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 $d$ .
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 $\theta = 78^\circ$ .

Substitute the values into the formula.
$d^2 = (7.5)^2 + (8.9)^2 - 2(7.5)(8.9) \cos(78^\circ)$
Calculate the squares of the sides.
$(7.5)^2 = 56.25$ $(8.9)^2 = 79.21$
Calculate the product $2ab$.
$2(7.5)(8.9) = 15 \times 8.9 = 133.5$
Find the value of $\cos(78^\circ)$ .
$\cos(78^\circ) \approx 0.20791169$
Substitute these values back into the equation for $d^2$ .
$d^2 = 56.25 + 79.21 - 133.5 \times 0.20791169$ $d^2 = 135.46 - 27.750259$ $d^2 = 107.709741$
Take the square root to find $d$ .
$d = \sqrt{107.709741}$ $d \approx 10.37833$
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.
$d = \boxed{\text{10.4 cm}}$

Answer