Question

question
in $\triangle klm$, $l = 550$ cm, $m = 870$ cm and $\angle k = 99^\circ$. find the length of $k$, to the nearest centimeter.

Explanation:

Step1: Identify the Law to Use

We have a triangle with two sides and the included angle (SAS), so we use the Law of Cosines. The Law of Cosines formula for side \( k \) (opposite angle \( K \)) is \( k^{2}=l^{2}+m^{2}-2lm\cos(K) \).

Step2: Substitute the Given Values

Given \( l = 550 \) cm, \( m = 870 \) cm, and \( K = 99^{\circ} \). First, calculate \( \cos(99^{\circ}) \). Using a calculator, \( \cos(99^{\circ})\approx - 0.1564 \).

Now substitute into the formula:
\[

$$\begin{align*} k^{2}&=550^{2}+870^{2}-2\times550\times870\times(- 0.1564)\\ &=302500 + 756900+2\times550\times870\times0.1564\\ &=1059400+2\times550\times870\times0.1564 \end{align*}$$

\]

Calculate \( 2\times550\times870 = 2\times478500=957000 \)

Then \( 957000\times0.1564 = 957000\times0.1564 = 149674.8 \)

So \( k^{2}=1059400 + 149674.8=1209074.8 \)

Step3: Find the Length of \( k \)

Take the square root of \( k^{2} \): \( k=\sqrt{1209074.8}\approx1099.58 \approx 1100 \) (to the nearest centimeter)

Answer:

The length of \( k \) is approximately \(\boxed{1100}\) centimeters.