Question

1. determine side m .

details

1. 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 \( a = 25.060\) cm, \( b = 17.180\) cm, and the included angle \( C = 70.34^\circ\). To find the third side \( m\) (let's call it \( c\) for the formula), we use the Law of Cosines: \( c^{2}=a^{2}+b^{2}-2ab\cos(C)\).

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, \( a^{2}=(25.060)^{2}=628.0036\), \( b^{2}=(17.180)^{2}=295.1524\), and \( 2ab = 2\times25.060\times17.180 = 862.3016\).

Now, \( c^{2}=628.0036 + 295.1524-862.3016\times0.3365\).

Calculate \( 862.3016\times0.3365\approx290.164\).

Then \( c^{2}=628.0036 + 295.1524 - 290.164=632.992\).

Step3: Find the Square Root

Take the square root of \( c^{2}\) to find \( c\): \( c=\sqrt{632.992}\approx25.16\) cm. Wait, let's recalculate more accurately.

Wait, maybe I made a mistake in the angle. Let's use more precise calculation for \( \cos(70.34^\circ)\). Let's convert \( 70.34^\circ\) to decimal degrees. Using calculator: \( \cos(70.34^\circ)=\cos(70 + 0.34^\circ)=\cos(70^\circ) \cos(0.34^\circ)-\sin(70^\circ)\sin(0.34^\circ)\). But easier to use calculator directly. Let's use a calculator: \( 70.34^\circ\), \( \cos(70.34)\approx\cos(70.34) \approx 0.33647\).

Then \( 2ab\cos C=2\times25.06\times17.18\times0.33647\).

First, \( 25.06\times17.18 = 25.06\times17 + 25.06\times0.18 = 426.02+4.5108 = 430.5308\). Then \( 2\times430.5308 = 861.0616\). Then \( 861.0616\times0.33647\approx861.0616\times0.33647\approx289.7\).

Then \( a^{2}+b^{2}=628.0036 + 295.1524 = 923.156\). Then \( c^{2}=923.156 - 289.7 = 633.456\). Then \( c=\sqrt{633.456}\approx25.17\) cm. Wait, maybe the initial assumption of the sides. Wait, the problem says "the opposite side measures m". Wait, maybe the angle is between the two sides? Wait, the diagram: the two sides are 25.060 and 17.180, and the included angle is 70.34 degrees. So Law of Cosines is correct. Wait, maybe I mixed up the sides. Wait, let's re-express: Let the triangle have sides \( a = 25.060\), \( b = 17.180\), included angle \( C = 70.34^\circ\), find side \( c = m\).

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

Calculating \( a^{2}=25.060^2 = 628.0036\), \( b^{2}=17.180^2 = 295.1524\), \( 2ab = 2\times25.060\times17.180 = 862.3016\), \( \cos(70.34^\circ)\approx\cos(70.34) \approx 0.3364\).

Then \( 2ab\cos C = 862.3016\times0.3364\approx289.9\).

Then \( c^{2}=628.0036 + 295.1524 - 289.9 = 633.256\). Then \( c=\sqrt{633.256}\approx25.16\) cm. So approximately 25.16 cm.

Wait, maybe the angle is not included? Wait, the problem says "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." Oh! Wait, maybe the angle is not included. So it's a triangle with two sides and a non-included angle? Wait, that would be Law of Sines, but Law of Sines: \( \frac{m}{\sin(70.34^\circ)}=\frac{25.060}{\sin B}=\frac{17.180}{\sin A}\). But we don't know the other angles. Wait, the diagram: the two sides are 25.060 and 17.180, and the angle between them is 70.34 degrees, so the included angle. So the initial approach is correct.

Wait, let's use more precise calculation. Let's use a calculator for \( \cos(70.34^\circ)\):

70.34 degrees. Convert to radians: \( 70.34\times\frac{\pi}{180}\approx1.2277\) radians. \( \cos(1.2277)\approx0.3364\).

Then \( 2ab\cos C = 2\times25.06\times17.18\times0.3364\).

25.06 * 17.18 = let's calculate 25 * 17.18 = 429.5, 0.06 * 17.18 = 1.0308, so total 429.5 + 1.0308 = 430.5308. Then 2 * 430.5308 = 861.0616. Then 861.0616 * 0.3364 = 861.0616 * 0.3 + 861.0616 * 0.03 + 861.0616 * 0.006 + 861.0616 * 0.0004 = 258.31848 + 25.831848 + 5.1663696 + 0.34442464 = 258.31848 + 25.831848 = 284.150328 + 5.1663696 = 289.3166976 + 0.34442464 = 289.66112224.

Then \( a^{2}+b^{2}=628.0036 + 295.1524 = 923.156\). Then \( c^{2}=923.156 - 289.66112224 = 633.49487776\). Then \( c=\sqrt{633.49487776}\approx25.17\) cm. So approximately 25.17 cm.

Answer:

\( \approx25.17\) cm (or more precise value depending on calculator precision)