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Determine whether the given information results in one, two, or three triangles. Solve for any triangles. a=6, c=4, C=10

User Dangerisgo
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1 Answer

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SOLUTION

Given the question in the question tab, the following are the solution steps to get the answer

Step 1: Write the given sides and angles


a=6,c=4,Step 2: With the information provided, we can find the size of the angle opposite the third side which we shall call side b, and then calculate the length of side b. We use the sine rule which states that[tex](a)/(\sin A)=(b)/(\sin B)=(c)/(\sin C)

By substitution, we have


\begin{gathered} (6)/(\sin A)=(4)/(\sin10) \\ By\text{ cross multiplication, we have} \\ 6*\sin 10=4*\sin A \\ \sin A=(6*\sin10)/(4) \\ A=\sin ^(-1)((6*\sin10)/(4)) \\ A=15.09808661 \\ A\approx15^(\circ) \\ \text{This makes Angle B to be} \\ B=180-10+15(sum\text{ of angles in a triangle equals 180)} \\ B=180-25 \\ B=155^(\circ) \end{gathered}

Step 3: To calculate length b using the sine rule;


\begin{gathered} (b)/(\sin B)=(c)/(\sin C) \\ (b)/(\sin155)=(4)/(\sin 10) \\ b=(4*\sin 155)/(\sin 10) \\ b=9.735046285 \\ b\approx9.7 \end{gathered}

Therefore the sides of the first triangle becomes 6, 9.7 and 10 units while it’s angles are 15,10 and 155 degrees.

Step 4: Determine if there is another triangle

Also, if two sides are given as 6 units and 4 units respectively, then the third side can be calculated by using the Pythagoras theorem and this immediately presumes that it’s a right angled triangle (one of the angles equals 90 degrees). The theorem states that;


hyp^2=adj^2+opp^2

Taking the other two as 6 and 4 units, the formula now becomes;


\begin{gathered} b^2=a^2+c^2 \\ b^2=6^2+4^2 \\ b^2=36+16 \\ b=\sqrt[]{42} \\ b=6.480740698 \\ b\approx6.48^(\circ) \end{gathered}

Therefore the sides of the second triangle becomes 6, 6.48 and 4 units with angles as 10, 80 and 90 degrees.

User Daniellee
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