The problem requires us to construct a triangle accurately using a ruler and protractor, given the lengths of two sides and the angle between them. We are given one side is 8 cm, another side is 7 cm, and the angle between those sides is 30 degrees. We need to find the length of the third side, labeled as 'b', and give the answer to one decimal place. Since we cannot use a calculator, we need to measure it.

1. Problem Description

The problem requires us to construct a triangle accurately using a ruler and protractor, given the lengths of two sides and the angle between them. We are given one side is 8 cm, another side is 7 cm, and the angle between those sides is 30 degrees. We need to find the length of the third side, labeled as 'b', and give the answer to one decimal place. Since we cannot use a calculator, we need to measure it.

2. Solution Steps

Since the instructions state we must draw the triangle and measure the length of side $b$, I will use the Law of Cosines to verify the measurement. However, without drawing the triangle, the solution must be derived mathematically.

The Law of Cosines states:

$c^2 = a^2 + b^2 - 2ab \cos(C)$

In our case, let $a = 8$ cm, $c = 7$ cm, and $C = 30^{\circ}$.

Then, we are looking for the value of side $b$.

Rearranging the formula:

$7^2 = 8^2 + b^2 - 2(8)(b)\cos(30^{\circ})$

$49 = 64 + b^2 - 16b \cos(30^{\circ})$

We know that $\cos(30^{\circ}) = \frac{\sqrt{3}}{2}$

$49 = 64 + b^2 - 16b (\frac{\sqrt{3}}{2})$

$49 = 64 + b^2 - 8\sqrt{3}b$

$0 = b^2 - 8\sqrt{3}b + 15$

Now we use the quadratic formula to solve for $b$.

$b = \frac{-B \pm \sqrt{B^2 - 4AC}}{2A}$

Here, $A = 1$, $B = -8\sqrt{3}$, $C = 15$.

$b = \frac{8\sqrt{3} \pm \sqrt{(-8\sqrt{3})^2 - 4(1)(15)}}{2(1)}$

$b = \frac{8\sqrt{3} \pm \sqrt{192 - 60}}{2}$

$b = \frac{8\sqrt{3} \pm \sqrt{132}}{2}$

$b = \frac{8\sqrt{3} \pm 2\sqrt{33}}{2}$

$b = 4\sqrt{3} \pm \sqrt{33}$

$\sqrt{3} \approx 1.732$

$\sqrt{33} \approx 5.745$

$b = 4(1.732) \pm 5.745$

$b = 6.928 \pm 5.745$

We have two possible solutions:

$b_1 = 6.928 + 5.745 = 12.673$

$b_2 = 6.928 - 5.745 = 1.183$

Since the sum of two sides of a triangle must be greater than the third side,

$8 + 7 > b$, $15 > b$.

$8 + b > 7$

$7 + b > 8$

Therefore, we cannot have the value of $b = 12.673$ since $7+8$ must be larger than $b$. The only possible answer is therefore $1.183$. Rounded to one decimal place, the answer is $1.2$ cm.

3. Final Answer

1.2 cm