Prove that there is no triangle for
A=60 degrees,
a=4 and
b= 14
if you could show your reasoning and working that would be great. thank you so much!
draw angle A and side b.
Now, if it were a right triangle, with b the hypotenuse, side a would be the altitude, of length 14 sin60° = 12.12
So, you see, a little side a of length 4 will not reach down the the base, AB.
If C is the right angle, that means that tanA = 6/14, but that does not make A=60°.
To prove that there is no triangle with the given properties, we can use the triangle inequality theorem.
The triangle inequality theorem states that the sum of the lengths of any two sides of a triangle must be greater than the length of the third side. In other words, for a triangle with sides a, b, and c, where c is the longest side, the following condition must be met: a + b > c.
In this case, we are given that angle A = 60 degrees, side a = 4, and side b = 14. Let's assume that side a is the longest side (so c = a).
According to the triangle inequality theorem, we have:
4 + 14 > a
18 > a
However, we are also given that angle A is 60 degrees, which means that side a is opposite to angle A. In a triangle, the side opposite an angle is always longer than the measure of the angle. Thus, a > 60.
Combining the two inequalities, we have:
18 > a > 60.
This is a contradiction, as a number cannot be simultaneously greater than 18 and less than 60. Therefore, no such triangle can exist with the given properties.
In summary, by applying the triangle inequality theorem and using the fact that the side opposite an angle is always longer than the measure of the angle, we have shown that there is no triangle with angle A = 60 degrees, side a = 4, and side b = 14.