Question

1. determine the length of

① a if b = 3 cm and c = 4 cm
② b if a = 13 cm and c = 12 cm
③ c if a = 80 mm and b = 12 m

Explanation:

Step1: Recall Pythagorean theorem

For a right - triangle with sides \(a\), \(b\) and hypotenuse \(c\), \(a^{2}+b^{2}=c^{2}\). We can re - arrange it to find the unknown side.

Step2: Find \(a\) when \(b = 3\mathrm{cm}\) and \(c = 4\mathrm{cm}\)

Using \(a=\sqrt{c^{2}-b^{2}}\), substitute \(b = 3\) and \(c = 4\). Then \(a=\sqrt{4^{2}-3^{2}}=\sqrt{16 - 9}=\sqrt{7}\mathrm{cm}\)

Step3: Find \(b\) when \(a = 13\mathrm{cm}\) and \(c = 12\mathrm{cm}\)

Using \(b=\sqrt{c^{2}-a^{2}}\), substitute \(a = 13\) and \(c = 12\). Then \(b=\sqrt{13^{2}-12^{2}}=\sqrt{169 - 144}=\sqrt{25}=5\mathrm{cm}\)

Step4: First, make the units consistent when \(a = 80\mathrm{mm}\) and \(b = 18\mathrm{mm}\)

Since \(1\mathrm{cm}=10\mathrm{mm}\), \(a = 8\mathrm{cm}\) and \(b = 1.8\mathrm{cm}\). Using \(c=\sqrt{a^{2}+b^{2}}\), substitute \(a = 8\) and \(b = 1.8\). Then \(c=\sqrt{8^{2}+1.8^{2}}=\sqrt{64 + 3.24}=\sqrt{67.24}=8.2\mathrm{cm}=82\mathrm{mm}\)

Answer:

a. \(\sqrt{7}\mathrm{cm}\)
b. \(5\mathrm{cm}\)
c. \(82\mathrm{mm}\)