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(1) The angles of a triangle measure 104^(@),51^(@), and 25^(@). The perimeter of the triangle is 10m. Find, rounded to 2 decimal places, the length of each side of the triangle. REVIEW SET 12A

(11) The angles of a triangle measure 104∘,51∘ 104^{\circ}, 51^{\circ} , and 25∘ 25^{\circ} . The perimeter of the triangle is 10 m 10 \mathrm{~m} . Find, rounded to 22 decimal places, the length of each side of the triangle.\newlineREVIEW SET 1212A

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Q. (11) The angles of a triangle measure 104∘,51∘ 104^{\circ}, 51^{\circ} , and 25∘ 25^{\circ} . The perimeter of the triangle is 10 m 10 \mathrm{~m} . Find, rounded to 22 decimal places, the length of each side of the triangle.\newlineREVIEW SET 1212A
  1. Verify Triangle Angles: Verify if the angles form a triangle.\newlineSum of angles in a triangle = 180∘180^\circ.\newline104∘+51∘+25∘=180∘104^\circ + 51^\circ + 25^\circ = 180^\circ.
  2. Use Law of Sines: Use the Law of Sines to find the sides.\newlineThe Law of Sines states: asin⁡(A)=bsin⁡(B)=csin⁡(C)=k\frac{a}{\sin(A)} = \frac{b}{\sin(B)} = \frac{c}{\sin(C)} = k (where kk is a constant).\newlineLet's denote the sides opposite to angles 104∘104^\circ, 51∘51^\circ, and 25∘25^\circ as aa, bb, and cc respectively.
  3. Calculate Constant kk: Calculate the constant kk using the perimeter.\newlinePerimeter =a+b+c=10 m= a + b + c = 10\,m.\newlineAssume k=10sin⁡(104°)+sin⁡(51°)+sin⁡(25°)k = \frac{10}{\sin(104°) + \sin(51°) + \sin(25°)}.\newlinek≈10(0.970+0.777+0.423)=102.170k \approx \frac{10}{(0.970 + 0.777 + 0.423)} = \frac{10}{2.170}.\newlinek≈4.61k \approx 4.61.
  4. Find Side Lengths: Find each side using the constant kk.a=k×sin⁡(104°)≈4.61×0.970≈4.47 ma = k \times \sin(104°) \approx 4.61 \times 0.970 \approx 4.47\,m.b=k×sin⁡(51°)≈4.61×0.777≈3.58 mb = k \times \sin(51°) \approx 4.61 \times 0.777 \approx 3.58\,m.c=k×sin⁡(25°)≈4.61×0.423≈1.95 mc = k \times \sin(25°) \approx 4.61 \times 0.423 \approx 1.95\,m.
  5. Check Perimeter: Check if the calculated sides add up to the perimeter.\newline4.47 m+3.58 m+1.95 m≈10 m4.47\,\text{m} + 3.58\,\text{m} + 1.95\,\text{m} \approx 10\,\text{m}.

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