The sides of the parallelogram are 5 and 26. A straight line, parallel to one side of the parallelogram

| June 15, 2021 | education

The sides of the parallelogram are 5 and 26. A straight line, parallel to one side of the parallelogram, splits it into two unequal similar parallelograms. Find the side lengths of these parallelograms.

In the parallelogram ABCD AB = CD = 5, BC = AD = 26. A) Draw line MK parallel to sides AB and CD. 1. Consider parallelograms ABMK and KMCD. If they are similar, then: AB / MK = BM / MC = MK / CD = AK / KD = 5/5 = 1 ⇒ they are equal, this option does not work. 1. Consider parallelograms ABMK and MCDK: AB / MC = BM / CD = MK / DK = AK / MK. Consider the ratio AB / MC = BM / CD: BC = BM + MC ⇒ BM = BC – MC = 26 – MC. Thus: 5 / MC = (26 – MC) / 5; 26 * MC – MC² = 25; MC² – 26 * MC + 25 = 0. D = 26² – 4 * 25 = 676 – 100 = 576. MC₁ = (26 – √576) / 2 = (26 – 24) / 2 = 2/2 = 1. MC₂ = (26 + √576) / 2 = (26 + 24) / 2 = 50/2 = 25. Let MC = 1, then BM = 26 – 1 = 25. Answer: in the parallelogram ABMK the sides are equal AB = MK = 5 , BM = AK = 25; in the parallelogram MCDK, the sides are MC = KD = 1, KM = CD = 5. B) Draw a line EF parallel to sides BC and AD. 1. Consider parallelograms AEFD and EBCD. If they are similar, then: AE / EB = EF / BC = FD / CF = AD / EF = 26/26 = 1 ⇒ they are equal, this option is not suitable. 1. Consider parallelograms AEFD and BCFE: AE / BC = EF / CF = FD / EF = AD / BE. Consider the ratio AE / BC = AD / BE: AB = AE + BE ⇒ AE = AB – BE = 5 – BE. Thus: (5 – BE) / 26 = 26 / BE; 676 = 5 * BE – BE²; BE² – 5 * BE + 676 = 0. D = 5² – 4 * 676 = 26 – 2704 = – 2678 <0 ⇒ no solutions.



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