Find the Answer to Ambigous Triangle Questions Involving Angles < 90°

[Aya]

I don't understand how to do these questions, involving angles less than 90 degrees. I know the "rules" you have to follow like if a<bsina there are two triangles etc. but what i don't understand is when there is a question, with no picture, how are you supposed to know witch one is the "a" and witch one it the "b".

For example
Determine the number of possible triangles that could be drawn with the given measurments

trianglePQR, where..

angleB=27
b=25cm
c=30 cm

Pleas help!

 

[HallsofIvy]

Surely the convention was explained to you? "C" is the angle opposite side "c", "B" is the angle opposite side "b", and "A" is the angle opposite side "a".

Given "angle B= 27 degrees, b= 25 cm., c= 30 cm., how would you draw the triangle? Here's what I would do. Draw a horizontal line and label it "c". Make it 30 cm long. Since angle C is opposite that, angle B and angle A are at the ends of that line. Choose whichever end you prefer, label it "B" and draw measure off a 27 degree angle. Of course, side "b" is opposite that so the side you just drawn is "a" and you don't know how long it will be- just extend it as long as you can. Now go to the other end of your horizontal line. That's, since its the only possibility left, angle "A" and the third side there is side "b". You know it's 25 cm long but you don't know the angle. That's what compasses are for- strike an arc of a circle with radius 25 cm. Where it crosses side "a" is the third vertex of the triangle- angle "C". There are 3 possiblities: that arc might cut side "a" twice, it might just touch it once, or it might no touch it at all. In the second possibility, just touching, side "a" would be tangent to the circle and angle "C" would have to be a right angle. That would be the case if side "b" were equal to length of c times sin(B)= 30 sin(27). If b is longer than that, there are two triangles, less than that, none.

 

[Architeuthis Dux]

I understand your confusion and frustration with these types of problems. It can be challenging to visualize and solve these triangle questions without a picture or diagram. However, there are a few steps you can follow to find the answer to ambiguous triangle questions involving angles less than 90 degrees.

First, it is important to remember that in a triangle, the sum of all angles is always 180 degrees. So, if you are given one angle, you can find the other two angles by subtracting the given angle from 180.

In the example you provided, we are given angle B as 27 degrees. This means that angle P and angle R must add up to 153 degrees (180-27=153). Now, we know that angle PQR is a triangle, so angle Q must also be less than 90 degrees. This narrows down the possible values for angle Q to be between 0 and 153 degrees.

Next, we can use the law of sines to find the possible values for sides a and c. The law of sines states that in any triangle, the ratio of the length of a side to the sine of its opposite angle is constant. So, we can set up the following equation: a/sin A = c/sin C, where A and C are the angles opposite sides a and c, respectively.

In our example, we know the length of side c is 30 cm and the measure of angle C is 153 degrees (180-27=153). So, we can plug in these values and solve for a. This will give us the possible values for side a, which can help us determine the number of possible triangles that can be drawn.

Additionally, you can also use the law of cosines to find the length of side b. The law of cosines states that in any triangle, the square of a side is equal to the sum of the squares of the other two sides minus twice the product of the two sides and the cosine of the included angle. So, we can set up the following equation: b^2 = a^2 + c^2 - 2ac cos B, where B is the angle between sides a and c.

Using this equation, we can solve for the possible values of side b, which can also help us determine the number of possible triangles.

In conclusion, while it may be challenging to solve ambiguous triangle questions without a diagram, using the laws of sines