The problem asks us to draw a triangle using a ruler and protractor based on the given information: one side is 10 cm, another side is 9 cm, and the angle between them is 30 degrees. Then, we need to measure the length of the third side, labeled as $a$, and provide the answer to one decimal place.

1. Problem Description

The problem asks us to draw a triangle using a ruler and protractor based on the given information: one side is 10 cm, another side is 9 cm, and the angle between them is 30 degrees. Then, we need to measure the length of the third side, labeled as $a$, and provide the answer to one decimal place.

2. Solution Steps

Since we cannot physically draw and measure, we will use the cosine rule to calculate the length of side $a$.

The cosine rule states that for a triangle with sides $a$, $b$, and $c$, and angle $C$ opposite side $c$:

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

In our problem, we can let $a$ be the unknown side, and we are given two sides (9 cm and 10 cm) and the angle between them (30 degrees).

So we want to find $a$, and we know two other sides, say $b=9$ and $c=10$, and the angle $A=30^\circ$ opposite side $a$. We use the formula:

$a^2 = b^2 + c^2 - 2bc \cos(A)$

Substituting the given values:

$a^2 = 9^2 + 10^2 - 2(9)(10) \cos(30^\circ)$

$a^2 = 81 + 100 - 180 \cos(30^\circ)$

Since $\cos(30^\circ) = \frac{\sqrt{3}}{2} \approx 0.866$,

$a^2 = 181 - 180 (0.866)$

$a^2 = 181 - 155.88$

$a^2 = 25.12$

$a = \sqrt{25.12}$

$a \approx 5.01198563 \dots$

Rounding to one decimal place, we get $a \approx 5.0$ cm.

3. Final Answer

5. 0 cm