Question

1. determine side m .
2. in a scalene triangle the measurement of an angle is 70.34 degrees, the opposite side measures m. the measurements of the other sides are 25.060 centimeters and 17.180 centimeters.

Explanation:

Step1: Identify the Law to Use

We have a triangle with two sides \( a = 25.060 \, \text{cm} \), \( b = 17.180 \, \text{cm} \) and the included angle \( C = 70.34^\circ \). To find the third side \( m \) (let's call it \( c \)), we use the Law of Cosines, which states that \( c^{2}=a^{2}+b^{2}-2ab\cos(C) \).

Step2: Substitute the Values

Substitute \( a = 25.060 \), \( b = 17.180 \), and \( C = 70.34^\circ \) into the formula. First, calculate \( \cos(70.34^\circ) \). Using a calculator, \( \cos(70.34^\circ)\approx0.3365 \).

Then, calculate \( a^{2}=(25.060)^{2}=628.0036 \), \( b^{2}=(17.180)^{2}=295.1524 \), and \( 2ab = 2\times25.060\times17.180 = 2\times430.5308 = 861.0616 \).

Now, substitute into the Law of Cosines:
\[ \begin{align*} c^{2}&=628.0036 + 295.1524- 861.0616\times0.3365\\ &=923.156 - 861.0616\times0.3365\\ \end{align*} \]
Calculate \( 861.0616\times0.3365\approx289.74 \)

Then \( c^{2}=923.156 - 289.74 = 633.416 \)

Step3: Find the Square Root

Take the square root of \( c^{2} \) to find \( c \) (which is \( m \)): \( m=\sqrt{633.416}\approx25.17 \, \text{cm} \) (we can also check with more precise calculation of \( \cos(70.34^\circ) \)).

Wait, let's recalculate with more precise \( \cos(70.34^\circ) \). Using a calculator, \( 70.34^\circ \) in radians is \( 70.34\times\frac{\pi}{180}\approx1.2277 \) radians. \( \cos(1.2277)\approx\cos(70.34^\circ)\approx0.33648 \)

Then \( 2ab\cos(C)=2\times25.060\times17.180\times0.33648 \)

First, \( 25.060\times17.180 = 430.5308 \)

Then \( 2\times430.5308 = 861.0616 \)

Then \( 861.0616\times0.33648\approx861.0616\times0.3365\approx289.74 \) (same as before)

\( a^{2}+b^{2}=25.060^{2}+17.180^{2}=628.0036 + 295.1524 = 923.156 \)

\( 923.156-289.74 = 633.416 \)

\( \sqrt{633.416}\approx25.17 \, \text{cm} \) (more accurately, using a calculator for square root: \( \sqrt{633.416}\approx25.17 \))

Wait, but let's check again. Maybe I made a mistake in the angle. Wait, the angle is included between the two sides? Wait, the problem says "the measurement of an angle is 70.34 degrees, the opposite side measures m". Wait, maybe the angle is not included? Wait, no, in the diagram, the angle is between the two sides of length 25.060 and 17.180? Wait, the diagram shows the two sides as 25.060 and 17.180 with the included angle 70.34 degrees, so the Law of Cosines is correct.

Wait, let's recalculate with more precise steps:

\( a = 25.060 \), \( b = 17.180 \), \( C = 70.34^\circ \)

Law of Cosines: \( c^{2}=a^{2}+b^{2}-2ab\cos C \)

\( a^{2}=25.060\times25.060 = 628.0036 \)

\( b^{2}=17.180\times17.180 = 295.1524 \)

\( 2ab = 2\times25.060\times17.180 = 861.0616 \)

\( \cos(70.34^\circ) \): let's use a calculator for more precision. 70.34 degrees. Let's convert to decimal degrees, it's 70 + 0.34 degrees. 0.34 degrees is 0.34*60 = 20.4 minutes, 0.4*60=24 seconds. So 70°20'24''.

Using a calculator, \( \cos(70.34^\circ) \approx \cos(70 + 0.34) = \cos(70^\circ) \cos(0.34^\circ)-\sin(70^\circ)\sin(0.34^\circ) \). But easier to use calculator: \( \cos(70.34) \approx 0.336478 \)

Then \( 2ab\cos(C)=861.0616\times0.336478\approx861.0616\times0.336478\approx289.73 \)

Then \( c^{2}=628.0036 + 295.1524 - 289.73 = 923.156 - 289.73 = 633.426 \)

\( \sqrt{633.426}\approx25.17 \, \text{cm} \) (rounded to four significant figures, since the given values have four significant figures: 25.060 (five), 17.180 (five), 70.34 (four). So the answer should be around 25.17 cm.

Wait, maybe I made a mistake in the Law of Cosines application. Wait, the Law of Cosines is \( c^{2}=a^{2}+b^{2}-2ab\cos(C) \), where \( C \) is the included angle between sides \( a \) and \( b \). So if the angle is included between 25.060 and 17.180, then yes, that's correct.

Alternatively, maybe the angle is not included? But the problem says "the measurement of an angle is 70.34 degrees, the opposite side measures m". Wait, if it's the opposite side, then maybe it's the Law of Sines? Wait, no, we have two sides and the included angle, so Law of Cosines is correct. Wait, if it's a side opposite the angle, then we need two sides and the included angle to find the third side. Wait, let's re-read the problem: "In a scalene triangle the measurement of an angle is 70.34 degrees, the opposite side measures m. The measurements of the other sides are 25.060 centimeters and 17.180 centimeters."

Ah! Wait, maybe the angle of 70.34 degrees is not the included angle, but an angle with opposite side m, and the other two sides are 25.060 and 17.180. Wait, that would be two sides and a non-included angle, but that's the ambiguous case (SSA). But the triangle is scalene, so maybe it's the included angle. Wait, the diagram shows the angle between the two sides (25.060 and 17.180) with the angle 70.34 degrees, and m is the third side. So the diagram confirms that the angle is included between the two sides, so Law of Cosines is correct.

So recalculating with more precision:

\( a = 25.060 \), \( b = 17.180 \), \( C = 70.34^\circ \)

\( c^{2}=25.060^{2}+17.180^{2}-2\times25.060\times17.180\times\cos(70.34^\circ) \)

Calculate each term:

\( 25.060^{2}=25.06\times25.06 = (25 + 0.06)^{2}=25^{2}+2\times25\times0.06 + 0.06^{2}=625 + 3 + 0.0036 = 628.0036 \)

\( 17.180^{2}=17.18\times17.18=(17 + 0.18)^{2}=17^{2}+2\times17\times0.18 + 0.18^{2}=289 + 6.12 + 0.0324 = 295.1524 \)

\( 2\times25.060\times17.180 = 2\times(25.06\times17.18) \)

Calculate \( 25.06\times17.18 \):

\( 25\times17.18 = 429.5 \)

\( 0.06\times17.18 = 1.0308 \)

So \( 25.06\times17.18 = 429.5 + 1.0308 = 430.5308 \)

Then \( 2\times430.5308 = 861.0616 \)

\( \cos(70.34^\circ) \): using a calculator, \( 70.34^\circ \) in radians is \( 70.34\times\frac{\pi}{180}\approx1.2277 \) radians. \( \cos(1.2277)\approx0.336478 \)

Then \( 861.0616\times0.336478\approx861.0616\times0.336478 \)

Calculate \( 861\times0.336478 = 861\times0.3 + 861\times0.03 + 861\times0.006478 = 258.3 + 25.83 + 5.577 = 289.707 \)

And \( 0.0616\times0.336478\approx0.0207 \)

So total \( 289.707 + 0.0207\approx289.7277 \)

Then \( c^{2}=628.0036 + 295.1524 - 289.7277 = 923.156 - 289.7277 = 633.4283 \)

\( \sqrt{633.4283}\approx25.17 \) (since \( 25.17^{2}=25^{2}+2\times25\times0.17 + 0.17^{2}=625 + 8.5 + 0.0289 = 633.5289 \), which is close to 633.4283, so maybe a more precise calculation:

\( 25.17^{2}=633.5289 \)

\( 25.16^{2}=(25.17 - 0.01)^{2}=25.17^{2}-2\times25.17\times0.01 + 0.01^{2}=633.5289 - 0.5034 + 0.0001 = 633.0256 \)

Our \( c^{2}=633.4283 \), which is between \( 25.16^{2} \) and \( 25.17^{2} \)

\( 633.4283 - 633.0256 = 0.4027 \)

\( 633.5289 - 633.0256 = 0.5033 \)

So the fraction is \( \frac{0.4027}{0.5033}\approx0.8 \)

So \( c\approx25.16 + 0.01\times0.8 = 25.168\approx25.17 \, \text{cm} \)

So the length of side \( m \) is approximately \( 25.17 \, \text{cm} \) (or more precisely, using a calculator for square root of 633.4283: \( \sqrt{633.4283}\approx25.17 \))

Answer:

\( \boxed{25.17 \, \text{cm}} \) (or depending on precision, maybe 25.17 cm)