Question

ij is a midsegment of \\( \triangle fgh \\). ij = 11, fh = 16, and gh = 20. find the perimeter of \\( \triangle ijh \\).
the perimeter of \\( \triangle ijh \\) is \\( \square \\).

Explanation:

Step1: Use the mid - segment theorem

The mid - segment theorem states that the length of a mid - segment of a triangle is half the length of the parallel side. Since \( IJ \) is a mid - segment of \( \triangle FGH \), \( FG = 2IJ \). Given \( IJ = 11 \), so \( FG=2\times11 = 22 \).

Step2: Calculate the perimeter of \( \triangle IJH \)

The perimeter of \( \triangle IJH \) is \( IJ+JH + IH \). We know that \( IJ = 11 \), \( JH=\frac{GH}{2}\) (by mid - segment theorem, since \( IJ \) is a mid - segment, the segment connecting the mid - points divides the other sides proportionally). Given \( GH = 20 \), so \( JH = 10 \). Also, \( IH=\frac{FH}{2}\), given \( FH = 16 \), so \( IH = 8 \).

Step3: Sum up the side lengths

The perimeter \( P=IJ + JH+IH=11 + 10+27=48 \).

Answer:

\( 48 \)