Question

put the steps in order: 30 - 60 - 90 #1
solve for x by putting the steps in order.
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$$\frac { 21 } { x } = \frac { \sqrt { 3 } } { 2 }$$
$$21 \cdot 2 = x \cdot \sqrt { 3 }$$
$$\frac { 42 } { \sqrt { 3 } } = x$$
$$42 = x \cdot \sqrt { 3 }$$
$$\frac { \text { medium } } { \text { long } } = \frac { \text { medium } } { \text { long } }$$
$$24.2 \text { units } = x$$

Explanation:

Step1: Use the property of 30 - 60 - 90 triangle

In a 30 - 60 - 90 triangle, the ratio of the sides is \(1:\sqrt{3}:2\). The formula for the ratio of the medium side to the long side is \(\frac{\text{medium}}{\text{long}}=\frac{\text{medium}}{\text{long}}\)

Step2: Set up the proportion

Given the medium side \(= 21\) and the long side \(=x\), we have \(\frac{21}{x}=\frac{\sqrt{3}}{2}\)

Step3: Cross - multiply

Cross - multiplying gives \(21\times2=x\times\sqrt{3}\)

Step4: Simplify the left - hand side

\(42 = x\times\sqrt{3}\)

Step5: Solve for \(x\)

\(x=\frac{42}{\sqrt{3}}\)

Step6: Rationalize the denominator

\(x=\frac{42\sqrt{3}}{3}=14\sqrt{3}\approx24.2\) (since \(\sqrt{3}\approx1.732\), \(14\times1.732 = 24.248\approx24.2\))

Answer:

\(\frac{\text{medium}}{\text{long}}=\frac{\text{medium}}{\text{long}}\), \(\frac{21}{x}=\frac{\sqrt{3}}{2}\), \(21\times2=x\times\sqrt{3}\), \(42 = x\times\sqrt{3}\), \(\frac{42}{\sqrt{3}}=x\), \(24.2\) units \(=x\)