Question

PLEASE HELP ME!!!! I can't seem to get this right!

Answer 1

Use the law of sines when you're given: (a) 2 angles, 1 side OR (b) 2 sides, non-included angle (aka an angle not created by those two sides)

Use the law of cosines when you're given: (a) 3 sides OR (b) 2 sides, included angle

Since you're given the measurements of two angles (A and C) and one side (a), you can solve the triangle using the law of sines.

Start by drawing the triangle. Remember that the uppercase letters are the angles and the lowercase letters are the length of the sides opposite the angle with the same letter (see picture - letters in blue are given, letters in green are what we're trying to find).

The law of sines says: 
$\frac{a}{sin(A)} = \frac{b}{sin(B)} = \frac{c}{sin(C)}$

1) You are told that angle A = 40°, angle C = 70°, and side a = 20. That means you can plug these values into $\frac{a}{sin(A)} = \frac{c}{sin(C)}$ (which we know is true because of the law of sines) to find the length of side c:
$\frac{a}{sin(A)} = \frac{c}{sin(C)}\\ \frac{20}{sin(40\°)} = \frac{c}{sin(70\°)}\\ c = \frac{20sin(70\°)}{sin(40\°)} \\ c \approx 29.238$

The length of side c is about 29.238.

2) Also remember that all the angles in a triangle add up to 180°. We know two of the angles, A and C, so subtract A and C from 180 to find the measure of angle B:
$\angle B = 180\° - 40\° - 70\° = 70\°$

The measure of angle B is 70°.

3) Now you can use the law of sines to find the length of side B. You can use $\frac{a}{sin(A)} = \frac{b}{sin(B)}$ or  $\frac{c}{sin(C)} = \frac{b}{sin(B)}$. I'll be using the first one:
$\frac{a}{sin(A)} = \frac{b}{sin(B)} \\ \frac{20}{sin(40\°)} = \frac{b}{sin(70\°)}\\ b = \frac{20sin(70\°)}{sin(40\°)}\\ b \approx 29.238$

The length of side b is also about 29.238.
You can also say b ≈ 29.238 without doing that math because triange ABC is an isosceles triangle since two angles (C and B) are the same, which makes their corresponding sides (c and b) the same! 

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Your answer: C) B = 70°, b = 29.2, c = 29.2

Answer 2

A. is the solution. B. Cannot be solution because there is no one side measurement that is higher than the others. The hypotenuse is always the longest side of the triangle. From this we can rule out C. as well. Finally, we can rule out D., because the angle measures do not add up to 180 degrees, as any real triangle does. Therefore, the answer must be A.