Question

For each, determine the number of unique triangles that can be made from the following information

. a. For ∆???,??̅̅̅̅ = 4?, ̅??̅̅̅ = 4.7?, and ∠? = 57°.

b. For ∆???,∠? = 50°, ̅??̅̅̅ = 62, and ??̅̅̅̅ = 80

Answer 1

1.

S 7 5구 R


Using sine rule:
$\\\sin \angle T = \frac{\overline{SR} \sin \angle R }{\overline{ST}}\\\\ \sin \angle T = \frac{4*\sin 57\degree}{4.7} = 0.713\\\\ {\color{Blue} \angle T = \sin ^{-1}0.713 = 45.5\degree}$

From sum rule of angle of triangle:
Sum of the angle = 180
$\\\angle R + \angle S +\angle T = 180\degree\\ 57\degree + \angle S + 45.5\degree = 180\degree\\ \angle S = 180\degree - 57\degree - 45.5\degree\\ \angle \mathrm{{\color{Blue} S=77.5\degree}}$

For, side ST using sine rule:
$\\RT = \frac{ST*\sin S}{\sin R} = \frac{4.7*\sin 77.5\degree}{\sin 57\degree} = \mathrm{{\color{Blue} 5.4}7}$

So, there is the one triangel possible, whose all dimentions are calculated as above.


2.
For ∆???,∠? = 50°, ̅??̅̅̅ = 62, and ??̅̅̅̅ = 80

D 62 म E F


Using sine rule:
$\sin E = \frac{DF \sin F}{DE} = \frac{80*\sin 50}{62} = \frac{80*0.766}{62}= \frac{61.28}{62} = 0.988$
$E = \sin^{-1}0.988 = 98.72\degree \;\;or\;\;81.28\degree$
There are two angle psossible for E, So, total numbe of triagle are 2.
1. Acute Scalene Triangle
2. Obtuse Scalene Triangle

We can also calculate the remainig side of the triangle. As in question is just to find the number of triangle, for good practice you should calculate the reamining part. Plaese commnet if you face any issue in the solution I will be happy to hlep you!