In Triangle DEF, DE = 8cm and Angle D= 60 degrees. When EF = 7 cm, how many triangles are possible?

two

If DF=4 we have a right triangle, with EF=4√3 = 6.92

Moving F a bit toward D will enable us to make EF a bit longer (7.00)

Moving F a bit away from D will allow for EF a bit longer (7.00)

To determine the number of possible triangles in Triangle DEF, we can use the triangle inequality theorem, which states that the sum of any two sides of a triangle must be greater than the third side.

Since DE = 8cm and EF = 7cm, we need to find the range of possible values for DF.

To find the minimum value for DF, we can subtract EF from DE: DF ≥ DE - EF = 8cm - 7cm = 1cm.

To find the maximum value for DF, we can add EF to DE: DF ≤ DE + EF = 8cm + 7cm = 15cm.

So, the possible range of values for DF is 1cm ≤ DF ≤ 15cm.

Now, we need to consider the given angle D = 60 degrees. To form a unique triangle, the side opposite to angle D must be within the range of values we obtained for DF.

Let's draw a line segment of length DF = 1cm, and another line segment of length DF = 15cm. From the vertex D, we draw an arc with a radius of 7cm to represent the possible positions of point F.

If the line segment of length DF = 1cm intersects the arc, then it forms a triangle. If the line segment of length DF = 15cm does not intersect the arc, it does not form a triangle. For any other value of DF within the range 1cm ≤ DF ≤ 15cm, both the arc and the line segment intersect, forming a unique triangle.

Therefore, if the line segment of length DF = 1cm intersects the arc and the line segment of length DF = 15cm does not intersect the arc, there are 1 possible triangle. Otherwise, there are 2 possible triangles.