Final answer:
The length of the mid-segment of the isosceles trapezoid klmn with equal bases KL and MN, and sides in the ratio LM:KN of 8:9, is found by solving for LM and KN using the given perimeter, and then averaging the lengths of bases KL and MN.
Step-by-step explanation:
The student's question involves finding the length of the mid-segment of a trapezoid, given certain conditions. Recall that in a trapezoid, the mid-segment connects the midpoints of the non-parallel sides and its length is equal to the average of the lengths of the two bases. As the trapezoid klmn has bases KL and MN that are equal in length, and the ratio of the lengths of the non-parallel sides LM and KN is 8:9, we can use the given perimeter to find the lengths of LM and KN initially.
Since get we are given that m<1 = m<2, we can infer that klmn is an isosceles trapezoid, which simplifies the problem significantly because we know both bases are equal and the non-base sides are equal to each other as well. The perimeter (pklmn) equation would be: 2*base + LM + KN = 132, and combined with the ratio LM/KN = 8/9, we can solve for LM and KN. Once those lengths are determined, the length of the mid-segment is simply the average of the two bases or simply the length of one of the bases as KL = MN.
Therefore, the answer would be equivalent to the length of base KL or MN, which is part of the trapezoid's bases, as the mid-segment of an isosceles trapezoid is equal to the lengths of its two equal bases.