Question

question
in δmno, m = 8.5 inches, m∠n = 52° and m∠o = 23°. find the length of o, to the nearest 10th of an inch.
answer attempt 1 out of 3
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Explanation:

Step1: Find the measure of angle M

In a triangle, the sum of interior angles is \(180^\circ\). So, \(m\angle M=180^\circ - m\angle N - m\angle O\).
Substitute \(m\angle N = 52^\circ\) and \(m\angle O = 23^\circ\):
\(m\angle M=180 - 52 - 23=105^\circ\)

Step2: Apply the Law of Sines

The Law of Sines states that \(\frac{o}{\sin M}=\frac{m}{\sin O}\). We know \(m = 8.5\) inches, \(m\angle M = 105^\circ\), \(m\angle O = 23^\circ\). We need to find \(o\).
Rearrange the formula for \(o\): \(o=\frac{m\times\sin M}{\sin O}\)

Step3: Calculate the sines of the angles

\(\sin105^\circ=\sin(60^\circ + 45^\circ)=\sin60^\circ\cos45^\circ+\cos60^\circ\sin45^\circ=\frac{\sqrt{3}}{2}\times\frac{\sqrt{2}}{2}+\frac{1}{2}\times\frac{\sqrt{2}}{2}=\frac{\sqrt{6}+\sqrt{2}}{4}\approx0.9659\)
\(\sin23^\circ\approx0.3907\)

Step4: Substitute the values and calculate \(o\)

Substitute \(m = 8.5\), \(\sin M\approx0.9659\), \(\sin O\approx0.3907\) into the formula for \(o\):
\(o=\frac{8.5\times0.9659}{0.3907}\)
First, calculate the numerator: \(8.5\times0.9659 = 8.21015\)
Then, divide by the denominator: \(o=\frac{8.21015}{0.3907}\approx21.0\) (Wait, no, wait. Wait, maybe I mixed up the angles. Wait, in triangle \(MNO\), the side \(m\) is opposite angle \(M\), side \(n\) opposite angle \(N\), side \(o\) opposite angle \(O\)? Wait, no, standard notation: in triangle \(ABC\), side \(a\) is opposite angle \(A\), side \(b\) opposite angle \(B\), side \(c\) opposite angle \(C\). So in triangle \(MNO\), side \(m\) is opposite angle \(M\), side \(n\) opposite angle \(N\), side \(o\) opposite angle \(O\). Wait, the problem says "Find the length of \(o\)". Wait, the given side is \(m = 8.5\) inches. So angle \(M\) is opposite side \(m\), angle \(N\) opposite side \(n\), angle \(O\) opposite side \(o\). Wait, maybe I made a mistake in the Law of Sines ratio. Let's re - check.

Law of Sines: \(\frac{m}{\sin M}=\frac{o}{\sin O}\) (since \(m\) is opposite \(\angle M\), \(o\) is opposite \(\angle O\)). Wait, no: \(\frac{\text{side opposite}\angle M}{\sin M}=\frac{\text{side opposite}\angle O}{\sin O}\). So side opposite \(\angle M\) is \(m\), side opposite \(\angle O\) is \(o\). So \(\frac{m}{\sin M}=\frac{o}{\sin O}\), so \(o=\frac{m\times\sin O}{\sin M}\). Oh! I had the ratio reversed earlier. That was the mistake.

Let's correct that.

Step3 (corrected): Apply the correct Law of Sines ratio

\(\frac{m}{\sin M}=\frac{o}{\sin O}\) implies \(o = \frac{m\times\sin O}{\sin M}\)

Step4 (corrected): Calculate the sines of the angles (same as before)

\(\sin M=\sin105^\circ\approx0.9659\), \(\sin O=\sin23^\circ\approx0.3907\), \(m = 8.5\)

Step5 (corrected): Substitute the values and calculate \(o\)

\(o=\frac{8.5\times0.3907}{0.9659}\)
First, calculate the numerator: \(8.5\times0.3907=3.32095\)
Then, divide by the denominator: \(o=\frac{3.32095}{0.9659}\approx3.4\)

Wait, that's different. Let's re - establish the triangle notation. Let's denote:

- Vertex \(M\), angle \(M\)
- Vertex \(N\), angle \(N = 52^\circ\)
- Vertex \(O\), angle \(O = 23^\circ\)

Side opposite angle \(M\): \(m\) (length 8.5 inches)

Side opposite angle \(O\): \(o\) (what we need to find)

So by Law of Sines: \(\frac{o}{\sin O}=\frac{m}{\sin M}\)

So \(o=\frac{m\times\sin O}{\sin M}\)

Now, \(\sin M=\sin(180 - 52 - 23)=\sin105^\circ\approx0.9659\)

\(\sin O=\sin23^\circ\approx0.3907\)

\(m = 8.5\)

So \(o=\frac{8.5\times0.3907}{0.9659}\)

\(8.5\times0.3907 = 3.32095\)

\(3.32095\div0.9659\approx3.4\)

Let's verify the angle sum: \(52 + 23+105 = 180\), which is correct.

So the correct calculation is \(o=\frac{8.5\times\sin23^\circ}{\sin105^\circ}\)

Calculating:

\(\sin23^\circ\approx0.3907\), \(\sin105^\circ\approx0.9659\)

\(8.5\times0.3907 = 3.32095\)

\(3.32095\div0.9659\approx3.4\)

Answer:

\(3.4\)