Question

1. find the length of the unmarked side in each right triangle. give your answers to one decimal place.

a) 3 cm, 6 cm
b) 4 cm, 12 cm
c) 2 cm, 10 cm
d) 8 cm, 4 cm

Explanation:

Step1: Recall Pythagorean theorem

For a right - triangle with sides \(a\), \(b\) and hypotenuse \(c\), \(a^{2}+b^{2}=c^{2}\). If we need to find a non - hypotenuse side, we can rewrite it as \(a=\sqrt{c^{2}-b^{2}}\) or \(b = \sqrt{c^{2}-a^{2}}\), and if we need to find the hypotenuse \(c=\sqrt{a^{2}+b^{2}}\).

Step2: Solve part a

Let \(a = 3\mathrm{cm}\) and \(b = 6\mathrm{cm}\). We find the hypotenuse \(c\). Using \(c=\sqrt{a^{2}+b^{2}}\), we have \(c=\sqrt{3^{2}+6^{2}}=\sqrt{9 + 36}=\sqrt{45}\approx6.7\mathrm{cm}\).

Step3: Solve part b

Let \(a = 4\mathrm{cm}\) and \(b = 12\mathrm{cm}\). We find the hypotenuse \(c\). Using \(c=\sqrt{a^{2}+b^{2}}\), we have \(c=\sqrt{4^{2}+12^{2}}=\sqrt{16+144}=\sqrt{160}\approx12.6\mathrm{cm}\).

Step4: Solve part c

Let the hypotenuse \(c = 10\mathrm{cm}\) and one side \(a = 2\mathrm{cm}\). We find the other side \(b\). Using \(b=\sqrt{c^{2}-a^{2}}\), we have \(b=\sqrt{10^{2}-2^{2}}=\sqrt{100 - 4}=\sqrt{96}\approx9.8\mathrm{cm}\).

Step5: Solve part d

Let the hypotenuse \(c = 8\mathrm{cm}\) and one side \(a = 4\mathrm{cm}\). We find the other side \(b\). Using \(b=\sqrt{c^{2}-a^{2}}\), we have \(b=\sqrt{8^{2}-4^{2}}=\sqrt{64 - 16}=\sqrt{48}\approx6.9\mathrm{cm}\).

Answer:

a) \(6.7\mathrm{cm}\)
b) \(12.6\mathrm{cm}\)
c) \(9.8\mathrm{cm}\)
d) \(6.9\mathrm{cm}\)