Question

A triangle has side lengths measuring 3x cm, 7x cm, and h cm. Which expression describes the possible values of h, in cm? 4x &lt; h < 10x 10x < h < 4x h = 4x h = 10x

Answer 1

we know that

The Triangle Inequality Theorem states that the sum of any 2 sides of a triangle must be greater than the measure of the third side

so

Three conditions mus be satisfied

Condition N $1$

$3x+7x > h\\ 10x > h\\h< 10x$

Condition N $2$

$3x+h > 7x\\ h > 7x-3x\\h> 4x$

Condition N $3$

$7x+h > 3x\\ h > 3x-7x\\h> -4x$ (condition N 3 is included in condition N 2)

therefore

$h> 4x\\ h< 10x\\ 4x < h < 10x$

the answer is the option

4x < h < 10x

Answer 2

Sum of two sides of triangle always greater than third side, so the possible values of h is:

7x-3x < h < 7x+3x
4x < h < 10x