in triangle qrs, qr = 8 and rs = 5. which expresses all possible lengths of side qs?\n○ qs = 13\n○ 5 < qs <…

in triangle qrs, qr = 8 and rs = 5. which expresses all possible lengths of side qs?\n○ qs = 13\n○ 5 < qs < 8\n○ qs > 13\n○ 3 < qs < 13

in triangle qrs, qr = 8 and rs = 5. which expresses all possible lengths of side qs?\n○ qs = 13\n○ 5 < qs < 8\n○ qs > 13\n○ 3 < qs < 13

Answer

Explanation:

Step1: Recall Triangle Inequality Theorem

The triangle inequality theorem states that the sum of the lengths of any two sides of a triangle must be greater than the length of the remaining side. For triangle ( QRS ) with sides ( QR = 8 ), ( RS = 5 ), and ( QS = x ) (let's denote ( QS ) as ( x )), we have three inequalities:

  1. ( QR + RS>QS )
  2. ( QR + QS>RS )
  3. ( RS + QS>QR )

Step2: Apply the inequalities

  • From ( QR + RS>QS ): ( 8 + 5>x ) which simplifies to ( x < 13 ).
  • From ( QR + QS>RS ): ( 8 + x>5 ). Subtracting 8 from both sides gives ( x>- 3 ). But since the length of a side cannot be negative, this inequality is not very restrictive here.
  • From ( RS + QS>QR ): ( 5 + x>8 ). Subtracting 5 from both sides gives ( x>3 ).

Combining the relevant inequalities (( x>3 ) and ( x < 13 )), we get ( 3<QS<13 ).

Answer:

( 3 < QS < 13 ) (the option with this expression)