Right angle triangle sides?

Question

I have a 30-60-90
A=90, B = 60, C= 30
If the side from A to C is 34"
How long is the length from C to B?

Answer 1

the way the angles are set up C to B is the hypotenuse


So Sin O/H
Cos A/H
Tan O/A

Cos(30) = 34/H
so
H=34/cos(30)

Or
Sin(60)=34/H
so
H=34/sin(60)

should be the same
39.26

Answer 2

There are some special characteristics about 30-60-90 triangles that are helpful to know in solving problems like this. In such a triangle, the side opposite the 30 degree angle is always half the length of the hypotenuse. That makes it easy to calculate the lengths of the other sides.

Suppose we let the side opposite the 30 degree angle have length 1. Then the hypotenuse has length 2 and the third side, the side opposite the 60 degree angle, has length sq rt (2² - 1²) = sq rt (3), by the Pythagorean Theorem. So, in any 30-60-90 triangle, the sides will always be in this ratio: 1, sq rt (3), 2, where the length of the sides can be paired with the angle to which they are opposite: (30, 1), (60, sq rt (3)), (90,2). Remembering these simple facts can make dealing with any 30-60-90 triangle a piece of cake in the future.

In your problem, the triangle is constructed in such a way that the side AC = 34" is opposite the 60 degree angle. From looking at the triangle, we can see that what we need to calculate is the sine of the 60 degree angle. We know that the sides which give the sine ratio (the side opposite the 60 degree angle divided by the hypotenuse) will be reducible to [sq rt (3) / 2]. We also know that the side opposite the 60 degree angle is 34". So all we have to do is set up an equation which will give us the correct ratio. When we do that, we will be finding the length of the hypotenuse, which is the side whose length we are interested in finding.

34 / x = sq rt (3) / 2 ---> x = [(34) * (2 / sq rt (3)] = [68 sq rt (3)] / 3.

So CB = [68 sq rt (3) / 3]", which is approximately equal to 39.26". That means the side opposite the 30 degree angle is half that length, [34 sq rt (3) / 3]", or about 19.63".

To check our answer, we simply plug them into the Pythagorean Formula and see whether it makes the equation true. If it does, our answer is correct.

x² = (34)² + (x/2)²
[68 sq rt (3) / 3] ² = (34)² + [34 sq rt (3) / 3]²
[(4624 * 3) / 9] = 1156 + [(1156 * 3) / 9]
(4624 / 3) = [(1156 * 3) / 3] + (1156 / 3)
(4624 / 3) = (3468 / 3) + (1156 / 3)
(4624 / 3) = [(3468 + 1156) / 3]
(4624 / 3) = (4624 / 3)

So, our calculated answer must be correct, and CB must equal [68 sq rt (3) / 3].

Answer 3

There is a cool, and easy way to do these....even tho Kate has a great answer and deserves the points!

My purpose is only to show you another way to arrive at answers to these kinds of problems.

A 30/60/90 (right) triangle has sides of 1, 2, and √3. Based on the triangle it's obvious which value goes to which side (2 being the longest, hence the hypotenuse)

Anyway, for your triangle it HAS to be this way:

90° lower right
60° lower left
30° upper

That's the orientation. Then AC is the vertical leg, which is 34".

So...the 'base' triangle would have sides:

√3 (vertical)
1 (lower leg)
2 (hypotenuse)

You need the multiplier for each leg. You know you have 34, and this side is √3 on the 'base' triangle:

34/√3 is the multiplier for each leg. Then the lower leg is 1(34/√3) and the hypotenuse is 2(34/√3), which is the side you want which gives 39.26".

It's MUCH faster and easier to do once you really see what's going on and I use this all the time for these standard triangles, it just looks more involved with all the 'explaining'. If you'd like more examples about this, let me know....send me a message. I'll be happy to explain this in more detail.

Glory be to Jesus indeed!

Answer 4

You have AC = 34, CB = x and AB = x/2 (because in a 30-60-90, the short side is 1/2 of the hypotenuse).

Plug these into the Pythagorean Theorem and you'll have an equation to solve.

Answer 5

BC / AC = 2 / √3 , for 30-60-90 special triangle, the ratio of the hypotenuse to the longer leg to is always 2 / √3.

BC = 34(2/√3) = 39.26 ''

Answer 6

you can use the sine rule, where:

sin60 = 34/CB
CB = 39.3 (68/cube root 3)

OR you can use cosine rule, where:

cos30 = 34/CB
CB = 39.3 (68/ cube root 3)

Answer 7

bc= root [4ac(sq)/3]
because of your angles ab =1/2bc, after that it's just Pitagora

Answer 8

cos 30 = AC/BC => BC=AC/cos 30 = 39.26 "