Question

in δhij, h = 340 cm, m∠j = 116° and m∠h = 5°. find the length of j, to the nearest 10th of a centimeter.

Explanation:

Step1: Find angle $I$

The sum of angles in a triangle is $180^{\circ}$. So $m\angle I=180^{\circ}-m\angle J - m\angle H$. Substituting the given values, $m\angle I = 180^{\circ}-116^{\circ}-5^{\circ}=59^{\circ}$.

Step2: Use the Law of Sines

The Law of Sines states that $\frac{h}{\sin H}=\frac{j}{\sin J}$. We know $h = 340$ cm, $m\angle H = 5^{\circ}$, and $m\angle J=116^{\circ}$. Rearranging for $j$, we get $j=\frac{h\sin J}{\sin H}$.

Step3: Calculate the value of $j$

Substitute the values into the formula: $j=\frac{340\times\sin(116^{\circ})}{\sin(5^{\circ})}$. We know that $\sin(116^{\circ})\approx0.8988$ and $\sin(5^{\circ})\approx0.0872$. Then $j=\frac{340\times0.8988}{0.0872}=\frac{305.592}{0.0872}\approx3504.5$.

Answer:

$3504.5$ cm