Question

In triangle $ABC,AB=8\text{units},AC=\text{units, and}BC=12\text{units If}BC$ is doubled, keeping the other two sides the same, then which of the following statements is true?

• A. According to the triangle inequality theorem, a unique triangle is possible.
• B. According to the triangle inequality theorem, no triangle is possible.
• C. According to the angle sum property, no triangle is possible.
• D. According to the angle sum property, a unique triangle is possible.

Answer 1

The correct option is B According to the triangle inequality theorem, no triangle is possible.

In triangle $ABC,BC$ is doubled. Let the new length be $B^{\prime }{C}^{\prime }$.
$B^{\prime }{C}^{\prime }=2\times BC$
$B^{\prime }{C}^{\prime }=2\times 12$
$B^{\prime }{C}^{\prime }=24\text{units}$
$AB=8\text{units},AC=5\text{units}$

According to the triangle inequality theorem, for any given triangle, the sum of the two sides of a triangle is always greater than the third side. To find whether $24$ is from the possible values of the third side of the triangle, $\text{we have to find the difference of the two sides < unknown side < sum of the two sides.}$

$8-5<x<8+5$ (Triangle inequality theorem)
$3<x<13,$
$24>13$

Hence, triangles cannot be formed with the given sides.