Question

Determine if there are zero, one, or two triangles for the following:

Answer 1

Given the data of the sides and the included angle, it can be said that there exist two triangles. Since a is greater than (b sin A), therefore there are two solutions or two triangles in the given measures.

Answer 2

Answer with explanation:

1.

In Δ ABC

 m ∠A=48°

a=10 m

b=12 m

Using Sine Law

$\frac{a}{\sin A}=\frac{b}{\sin B}=\frac{c}{\sin C}\\\\\frac{10}{\sin 48^{\circ}}=\frac{12}{\sin B}\\\\ \ sinB^{\circ}=\frac{12 \times \sin 48^{\circ}}{10}\\\\\ sinB^{\circ}=\frac{12 \times 0.7431}{10}\\\\\ sinB^{\circ}=0.9\\\\B=65^{\circ}$

So, Angle C can be obtained by using angle sum property of triangle.

So,This is one kind of triangle, having measure of three Angles are , 48°, 65°,67°.

2.

≡≡Now, we will use cosine law to find if there is another triangle having measure of one angle 48°.

$\cos 48^{\circ}=\frac{12^2+c^2-10^2}{2 \times 12 \times c}\\\\0.67=\frac{144+c^2-100}{24 c}\\\\16.08 c=44+c^2\\\\100 c^2 -1608 c +4400=0\\\\25 c^2-402 c+1100=0\\\\c=12.58\\\\c=3.50$

Sum of two sides of triangle should always be greater than third side.

So, c=3.50

and, c= 12.58

Triangle of two type is possible having dimension, a=10 m, b=12 m, c=12.58 m and  a=10 m, b=12 m, c=3.50 m.

But,when side c=12.58 m, then ∠C=67°.

So, There are two triangles possible in this case,out of which one is Congruent with triangle obtained in case 1.

So,There are two distinct Triangles.

⇒⇒⇒Two Triangles