Question

two sides and one angle measure of the triangle shown are known. which best describes a method to derive a formula for the area of the triangle using the given information?
draw an auxiliary line from vertex q that bisects lp. use the cosine formula to determine that the length of the auxiliary line is 18(cos87°). multiply this by 1/2 the length of lp, 16 inches.
draw an auxiliary line from vertex q that bisects lp. use the sine formula to determine that the length of the auxiliary line is 18(sin87°). multiply this by 1/2 the length of lp, 16 inches.
draw an auxiliary line from vertex q that is perpendicular to lp. use the sine formula to determine that the length of the auxiliary line is 18(sin87°). multiply this by 1/2 the length of lp, 16 inches.
draw an auxiliary line from vertex q that is perpendicular to lp. use the cosine formula to determine that the length of the auxiliary line is 18(cos87°). multiply this by 1/2 the length of lp, 16 inches.

Explanation:

Step1: Recall the formula for the area of a triangle

The formula for the area of a triangle is \(A=\frac{1}{2}\times base\times height\). Here, if we consider \(LP\) as the base (\(b = 16\) in), we need to find the height (\(h\)) relative to this base.

Step2: Analyze the construction of the height

To find the height relative to the base \(LP\), we draw an auxiliary line from vertex \(Q\) that is perpendicular to \(LP\). Let's call the height \(h\). If we consider the side \(QL = 18\) in and the angle \(\angle L=87^{\circ}\), using the sine function in a right - triangle (formed by the height, a part of \(LP\), and the side \(QL\)). In a right - triangle, \(\sin\theta=\frac{\text{opposite}}{\text{hypotenuse}}\). Here, \(\sin(87^{\circ})=\frac{h}{18}\), so \(h = 18\sin(87^{\circ})\)

Step3: Calculate the area

Substitute \(b = 16\) and \(h=18\sin(87^{\circ})\) into the area formula \(A=\frac{1}{2}\times b\times h\). So \(A=\frac{1}{2}\times16\times18\sin(87^{\circ})\)

Answer:

Draw an auxiliary line from vertex \(Q\) that is perpendicular to \(\overline{LP}\). Use the sine formula to determine that the length of the auxiliary line is \(18(\sin87^{\circ})\). Multiply this by \(\frac{1}{2}\) the length of \(LP\), \(16\) inches.