Question

in △pqr, the obtuse - angle is 105°, and the shorter sides measure 4 and 7 units. rounded to the nearest tenth, what is the area of △pqr? area = square units

Explanation:

Step1: Recall area formula for a triangle

The area formula for a triangle with two - side lengths \(a\) and \(b\) and included - angle \(C\) is \(A=\frac{1}{2}ab\sin C\).

Step2: Identify values of \(a\), \(b\), and \(C\)

Let \(a = 4\), \(b = 7\), and \(C=105^{\circ}\). We know that \(\sin(105^{\circ})=\sin(60^{\circ}+45^{\circ})\). Using the sum - formula for sine \(\sin(A + B)=\sin A\cos B+\cos A\sin B\), \(\sin(60^{\circ}+45^{\circ})=\sin60^{\circ}\cos45^{\circ}+\cos60^{\circ}\sin45^{\circ}=\frac{\sqrt{3}}{2}\times\frac{\sqrt{2}}{2}+\frac{1}{2}\times\frac{\sqrt{2}}{2}=\frac{\sqrt{6}+\sqrt{2}}{4}\approx\frac{2.449 + 1.414}{4}=\frac{3.863}{4}=0.966\).

Step3: Calculate the area

Substitute \(a = 4\), \(b = 7\), and \(\sin C\approx0.966\) into the area formula \(A=\frac{1}{2}ab\sin C\). So \(A=\frac{1}{2}\times4\times7\times0.966\). First, \(\frac{1}{2}\times4\times7 = 14\). Then \(A = 14\times0.966=13.524\approx13.5\).

Answer:

13.5