Question

Solve for the length of PQ in the following triangle.

A) 2x+6
B) 5x−8
C) x+6
D) 3x+8

Answer 1

The length of PQ in the given triangle is represented by option C) x+6.

Explanation:

In the given triangle, let's denote the length of PQ as (PQ = a). According to the triangle inequality theorem, the sum of the lengths of any two sides of a triangle must be greater than the length of the third side. Applying this theorem to our triangle with sides represented by expressions, we get:

`\[2x + 6 + 5x - 8 > x + 6\]`

Combining like terms:

`\[7x - 2 > x + 6\]`

Subtracting (x) from both sides:

`\[6x - 2 > 6\]`

Adding 2 to both sides:

`\[6x > 8\]`

Dividing both sides by 6:

`\[x > (4)/(3)\]`

Since (x) must be greater than
`\((4)/(3)\)` for the inequality to hold, we can choose option (x+6) as the expression for the length of PQ. This ensures that the sum of the lengths of the other two sides is always greater than the length of PQ, satisfying the triangle inequality theorem.

Therefore, the length of PQ is accurately represented by (x + 6), making option C) x+6 the correct choice.