Question

Two triangles share a side of length x
. The first triangle has side lengths 2, 9, and x
. The second triangle has side lengths 2, 7, and x
. Is 9 a possible length for x
? help

Answer 1

To determine if 9 is a possible length for x, we need to check if it satisfies the triangle inequality for both triangles.

For the first triangle, the triangle inequality states that the sum of any two side lengths of a triangle must be greater than the third side length. So, for the first triangle with side lengths 2, 9, and x, we need to check if 2 + 9 > x, 2 + x > 9, and 9 + x > 2.

Checking the first inequality, 2 + 9 > x, we have 11 > x.
Checking the second inequality, 2 + x > 9, we have x > 7.
Checking the third inequality, 9 + x > 2, we have 9 + x > 2, which is always true.

So, for the first triangle, we have the inequalities 11 > x and x > 7.

For the second triangle, the same triangle inequality needs to be satisfied. So, for the second triangle with side lengths 2, 7, and x, we need to check if 2 + 7 > x, 2 + x > 7, and 7 + x > 2.

Checking the first inequality, 2 + 7 > x, we have 9 > x.
Checking the second inequality, 2 + x > 7, we have x > 5.
Checking the third inequality, 7 + x > 2, we have 7 + x > 2, which is always true.

So, for the second triangle, we have the inequalities 9 > x and x > 5.

Combining both sets of inequalities, we get 11 > x > 9 and x > 7 > 5.

From these inequalities, we see that x must be greater than 9 and greater than 7, which means the possible values for x are x > 9. Since 9 is not greater than 9, it is not a possible length for x in this case.