Question

triangle area formula
in $delta pqr$, the obtuse angle is $105^circ$, and the shorter sides measure 4 and 7 units.
rounded to the nearest tenth, what is the area of $delta pqr$
area = $square$ square units

Explanation:

Step1: Recall triangle area formula

For two sides $a, b$ and included angle $\theta$, area $A=\frac{1}{2}ab\sin\theta$

Step2: Substitute given values

$a=4$, $b=7$, $\theta=105^\circ$, so $A=\frac{1}{2}\times4\times7\times\sin(105^\circ)$

Step3: Calculate $\sin(105^\circ)$

$\sin(105^\circ)=\sin(60^\circ+45^\circ)=\sin60^\circ\cos45^\circ+\cos60^\circ\sin45^\circ=\frac{\sqrt{3}}{2}\times\frac{\sqrt{2}}{2}+\frac{1}{2}\times\frac{\sqrt{2}}{2}=\frac{\sqrt{6}+\sqrt{2}}{4}\approx0.9659$

Step4: Compute final area

$A=\frac{1}{2}\times4\times7\times0.9659=14\times0.9659\approx13.5$

Answer:

13.5 square units