Answer: The value of the ratio is 3
This can be expressed as 3:1
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Explanation:
Draw out the right triangle ABC where the right angle is at angle C. Draw a perpendicular line segment from point C to the hypotenuse AB. Mark the angle BAC to be 30 degrees. Let the opposite side of the 30 degree angle be x units long. Let AH = y and HB = z. The goal is to find y/z which is the ratio AH:HB.
See Figure 1 (attached) for what the drawing should look like.
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Now add in the other angles such as ABC = 60 degrees. If you focus on triangle AHC, we have a right angle at angle H. At angle A we have the initial 30 degree angle. At angle C we have 60 degrees. See Figure 2.
Similarly, we have another 30-60-90 triangle for triangle CHB as shown in Figure 2.
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We'll use the special 30-60-90 triangle template as shown in Figure 3. Compare Figure 2 and Figure 3. Do this for triangle ABC. We see that AH+HB = y+z matches up with AB = 2x. This must mean that 2x = y+z
Furthermore, we have two additional 30-60-90 triangles which are smaller. For triangle AHC we have what you see in Figure 4. It looks a bit ugly but basically AH is equal to (x*sqrt(3))/2 since we divide the original x*sqrt(3) over 2. This is due to the fact that the hypotenuse is twice as large as the short leg. Now multiply the length of CH by value sqrt(3) to get what you see shown as the work on Figure 4. We end up wih y = (3x)/2. We'll use this later.
So write y = (3x)/2 down and put a box around it to remember to use it later.
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Now turn to Figure 5. This figure is a lot nicer than figure 4. Start with focusing on triangle CHB. The hypotenuse is BC = x. Half of this is x/2 which is the length of the short leg HB.
So z = x/2
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We found earlier that
y = (3x)/2
z = x/2
Divide the values to get
y/z = [ (3x)/2 ] divided by (x/2)
y/z = [ (3x)/2 ] * (2/x)
y/z = (3x*2)/(2*x)
y/z = (6x)/(2x)
y/z = 3
So after all that complicated messy work, we end up with the simple final answer of 3.
This can be expressed as 3:1
This means that AH is three times longer compared to HB (eg: AH could be 12 units long while HB is 4 units long)
