Question

4 - 1 practice
geometry (s1 p01 rm h218) sharp, jeffrey / unit 4 - right triangle trigonometry / lesson 4 - 1 angles and steepness

1. find the missing side in each triangle using any method. check your answers using a different method.

ac = units
de = units

Explanation:

Step1: Find \(AC\) using Pythagorean theorem

In right - triangle \(ABC\), by the Pythagorean theorem \(a^{2}+b^{2}=c^{2}\), where \(a = 5\), \(b\) is the unknown side, and \(c=12\). Wait, no, in right - triangle \(ABC\) with legs \(AB = 5\) and \(BC\) (opposite to the right angle \(B\)), and hypotenuse \(AC\). The Pythagorean theorem is \(AB^{2}+BC^{2}=AC^{2}\). Substituting \(AB = 5\) and \(BC = 12\), we get \(AC=\sqrt{5^{2}+12^{2}}=\sqrt{25 + 144}=\sqrt{169}=13\)

Step2: Find \(DE\) using Pythagorean theorem

In right - triangle \(DEF\) with legs \(DE\) and \(EF=10\), and hypotenuse \(DF = 26\). By the Pythagorean theorem \(DE^{2}+EF^{2}=DF^{2}\). Let \(DE=x\), then \(x^{2}+10^{2}=26^{2}\). So \(x^{2}=26^{2}-10^{2}=(26 + 10)(26 - 10)\) (using \(a^{2}-b^{2}=(a + b)(a - b)\)). \(x^{2}=(36)(16)=576\), and \(x=\sqrt{576}=24\)

Step3: Check \(AC\) using similarity (if \(\triangle ABC\sim\triangle DEF\))

The ratio of sides: \(\frac{AB}{EF}=\frac{5}{10}=\frac{1}{2}\), \(\frac{AC}{DF}=\frac{13}{26}=\frac{1}{2}\). Also, \(\angle B=\angle E = 90^{\circ}\) and if \(\angle A=\angle F\) (by AA similarity, since right angles and the ratio of sides). The Pythagorean result is consistent with similarity ratio.

Step4: Check \(DE\) using Pythagorean theorem again (as a simple check)

\(24^{2}+10^{2}=576+100 = 676\) and \(26^{2}=676\)

Answer:

\(AC = 13\) units, \(DE=24\) units