7.
Diagram NOT accurately drawn
\[
\begin{array}{l}
A B=3.2 \mathrm{~cm} \\
B C=8.4 \mathrm{~cm}
\end{array}
\]
The area of triangle $A B C$ is $10 \mathrm{~cm}^{2}$.
Calculate the perimeter of triangle $A B C$.
Give your answer correct to three significant figures.

Step 1 ：We are given that the lengths of sides AB and BC of triangle ABC are 3.2 cm and 8.4 cm respectively. The area of the triangle is given as 10 cm².

Step 2 ：We can use the formula for the area of a triangle, which is \(\frac{1}{2} \times \text{base} \times \text{height}\), to find the height of the triangle. In this case, the base is BC and the area is given, so we can solve for the height.

Step 3 ：Substituting the given values into the formula, we get \(10 = \frac{1}{2} \times 8.4 \times \text{height}\). Solving for height, we get \(\text{height} = \frac{10}{0.5 \times 8.4} = 2.380952380952381\) cm.

Step 4 ：Once we have the height, we can use the Pythagorean theorem to find the length of AC. The Pythagorean theorem states that in a right triangle, the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the lengths of the other two sides. We can use this theorem because we can consider triangle ABC as a right triangle with AB as one side, the height as the other side, and AC as the hypotenuse.

Step 5 ：Substituting the given values into the Pythagorean theorem, we get \(AC = \sqrt{(3.2)^2 + (2.380952380952381)^2} = 3.988600536574553\) cm.

Step 6 ：Finally, we can add up the lengths of AB, BC, and AC to find the perimeter of the triangle. The perimeter is given by \(AB + BC + AC = 3.2 + 8.4 + 3.988600536574553 = 15.588600536574555\) cm.

Step 7 ：Rounding to three significant figures, the perimeter of the triangle ABC is approximately \(\boxed{15.589}\) cm.