Question

In triangle ABD
𝐴
𝐵
𝐷
, AB
𝐴
𝐵
is 3 cm
3

cm
long and BD
𝐵
𝐷
is 5 cm
5

cm
long. According to the Triangle Inequality Theorem, which two lengths are possible lengths of AD
𝐴
𝐷
?

Answer 1

wrong

Answer 2

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.

In triangle \( ABD \), we know the lengths of sides \( AB \) and \( BD \):

- \( AB = 3 \) cm
- \( BD = 5 \) cm
- Let \( AD = x \) cm (the length we need to find).

According to the Triangle Inequality Theorem, we can write the following inequalities:

1. \( AB + BD > AD \):
\[
3 + 5 > x \implies 8 > x \implies x < 8
\]

2. \( AB + AD > BD \):
\[
3 + x > 5 \implies x > 5 - 3 \implies x > 2
\]

3. \( BD + AD > AB \):
\[
5 + x > 3 \implies x > 3 - 5 \implies x > -2
\]
(This inequality is always satisfied since \( x \) is positive.)

Combining the valid inequalities gives us:
\[
2 < x < 8
\]

Therefore, the possible lengths of \( AD \) (denoted as \( x \)) must be greater than 2 cm and less than 8 cm. This means the possible lengths of \( AD \) can lie in the range \( (2, 8) \).