Pythagoras to calculate the lengths of the unknown sides in the triangles. Leave irrational answers in the simplest surd form: 5.1 In \( \triangle \mathrm{ABC}, \hat{\mathrm{B}}=90^{\circ}, \mathrm{AC}=8 \mathrm{~cm}, \mathrm{BC}=4 \mathrm{~cm} \). Calculate the ler AB without using a calculator.

In triangle \( \triangle ABC \), where \( \angle B = 90^\circ \), we can use the Pythagorean theorem to find the length of side \( AB \). The Pythagorean theorem states that: \[ AC^2 = AB^2 + BC^2 \] Given: - \( AC = 8 \, \text{cm} \) - \( BC = 4 \, \text{cm} \) We can substitute these values into the equation: \[ 8^2 = AB^2 + 4^2 \] Calculating the squares: \[ 64 = AB^2 + 16 \] Now, we can isolate \( AB^2 \): \[ AB^2 = 64 - 16 \] \[ AB^2 = 48 \] To find \( AB \), we take the square root of both sides: \[ AB = \sqrt{48} \] We can simplify \( \sqrt{48} \): \[ \sqrt{48} = \sqrt{16 \times 3} = \sqrt{16} \times \sqrt{3} = 4\sqrt{3} \] Thus, the length of side \( AB \) is: \[ AB = 4\sqrt{3} \, \text{cm} \]