Given the following conditions, describe how to determine whether or not a unique triangle is formed.

Question

Side: 5 cm

Side: 6 cm

Non-included angle: 30 degrees

Answer 1

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Answer 2

You should draw the 6 cm side as a horizontal line segment, and attach a 'swinging' 5 cm side to the right endpoint of the horizontal side.

 

For a RIGHT triangle with base 6 cm & 30^o angle at the LEFT endpoint, the short leg will be of length 6/√3 which is about 3.464 cm*. Since the length 5 cm exceeds the cited height=3.346 cm, said 'SSA' triangle can be formed.  Imagine an arc of a circle of radius 5, which will intersect the (extended) hypotenuse of the right triangle at two points--so there are two possible triangles.

 

 

Answer 3

Emma,

 

If you attempt to sketch this triangle, put the third side on the bottom, the 6 cm side extending upward from this side and making the 30° angle with it, and the 5 cm side extending from the end of the 6 cm side of the angle downward.

 

If a triangle can be formed, the 5 cm side should be able to touch (intersect) the third side in at least one point. (This is hard to describe without being able to draw a picture, but you will need to try visualizing it.)

 

This can happen only when the 5 cm side is at least 6 sin 30° in length. This is the altitude of any triangle formed by the 30° angle and the 6 cm side. Since this equals 3 cm, and 5 cm is longer, a triangle is formed.

 

In fact, since 3 < 5 < 6, there are actually two triangles formed, with the 5 cm side touching the bottom side at two different points and forming two triangle, one obtuse and one acute. (Think of the 5 cm side swinging from the shared vertex with the 6 cm side, and you will see it touch the base in two different points.)

 

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