Question

Which three lengths could be the lengths of the sides of a triangle?

A) 6 cm, 23 cm, 11 cm
B) 10 cm, 15 cm, 24 cm
C) 22 cm, 6 cm, 6 cm
D) 15 cm, 9 cm, 24 cm

Answer 1

D
the answer is D because it is equilateral triangle

Answer 2

B.
`10` cm,
`15` cm,
`24` cm

Explanation:

we know that

The Triangle Inequality Theorem, states that the sum of the lengths of two sides of a triangle must always be greater than the length of the third side

so

`a+b > c`

`a+c > b`

`b+c >a`

where

a,b,c are the lengths sides of the triangle

case A)
`6` cm,
`23` cm,
`11` cm

Let

`a=6\ cm`

`b=23\ cm`

`c=11\ cm`

Verify

`a+b > c` ------>
`6+23 > 11` ------> is true

`a+c > b` ------>
`6+11 > 23` -------> is not true

therefore

The three lengths of case A) could not be the lengths of the sides of a triangle

case B)
`10` cm,
`15` cm,
`24` cm

Let

`a=10\ cm`

`b=15\ cm`

`c=24\ cm`

Verify

`a+b > c` ------>
`10+15 > 24` ------> is true

`a+c > b` ------>
`10+24 > 15` -------> is true

`b+c >a` ----->
`15+24 >10` --------> is true

therefore

The three lengths of case B) could be the lengths of the sides of a triangle

case C)
`22` cm,
`6` cm,
`6` cm

Let

`a=22\ cm`

`b=6\ cm`

`c=6\ cm`

Verify

`a+b > c` ------>
`22+6 > 6` ------> is true

`a+c > b` ------>
`22+6> 6` -------> is true

`b+c >a` ----->
`6+6 >22` --------> is not true

therefore

The three lengths of case C) could not be the lengths of the sides of a triangle

case D)
`15` cm,
`9` cm,
`24` cm

Let

`a=15\ cm`

`b=9\ cm`

`c=24\ cm`

Verify

`a+b > c` ------>
`15+9 > 24` ------> is not true

therefore

The three lengths of case D) could not be the lengths of the sides of a triangle