help with geometry?

Question

In trianlge ABC, angle A= 30, angle B= 60,
and BC = 4. What is the length of AC to the nearest hundredth?

If the length of an altitude of an equilater triangle is 5 root three, then the lenght of a side is
a) 4 b) 5 c) 5 root 3 d) 10 e) 10 root 3

Answer 1

AC is equal to 4*sqrt(3) or approx. = 4* 1.732. This is because in a 30-60-90 triangle, the ratio of side length is 1-2-sqrt of 3. The side opposite the 30 angle is 1, the side opposite the 60 angle is the square root of 3 and the side opposite the 90, or the hypotenuse, is 2. So the 4 (BC) is the 1, and AC is the square root of 3 (or four times that in this case).

Additionally then, the sine of angle A can be seen to be 0.5, since the sine of an angle is equal to the opposite/hypotenuse. The opposite side is 1, the hypotenuse is 2, and the sine of 30 is 1/2 = 0.5. Just another way to check your 30-60-90's.

For the second question, you can again use this, since dividing an equilat tri by an altitude is like making two 30-60-90's. Since the 5root3 is the side opposite a 60 (any of the three angles in an equilat), the small side of one of the 30-60-90's is 5, and since the equilat tri is made up of two of these, the side is 10, so answer d.

Answer 2

a triangle with angles 30,60,or 45 are special triangles. They have their own properties.

the opposite side of angle 30degrees has a value of a. then, the opposite side of the angle 60degrees has a value of a*sqrt3. and the last side of the triangle would have the value of 2a.
because A is 30degrees, BC would have the value of a, then a would be equal to 4. AC then would be equal to 4*sqrt3.
4*sqrt3 would be approximately be 6.93

5 root 3 would be the side opposite 60 degrees,so a=5
the other length would be 5 root 3. SO: c is the answer

Answer 3

4/sin30=AC/sin60

AC=8 sqrt(3)/2
=6,93



sin60=h/a
h=asin60
5sqrt(3)=a sqrt(3)/2

5=a/2
a=10

answer:
d) 10