Дано abcd трапеция , BM=MK=KE=EA, MN||KP||EF||AD,BC=14, AD=18. найти MN,KP,EF

Question 1

Задача Задача №38535

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Аноним

09 ноября 2023 06:50 · публичный вопрос

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Аноним 09 ноября 2023 06:50

Дано abcd трапеция , BM=MK=KE=EA, MN||KP||EF||AD,BC=14, AD=18. найти MN,KP,EF

Answer

To find the lengths of MN, KP, and EF in the trapezoid ABCD, we need to use the given information and the properties of trapezoids.

First, let's draw a diagram to visualize the trapezoid:

  A ________B
   /        \
  /          \
 D________C

Given:

BM = MK = KE = EA (the trapezoid ABCD is isosceles)
MN || KP || EF || AD (the sides MN, KP, EF, and AD are parallel)

To find MN: Since MN || KP, we can consider triangles BKM and KPC to be similar. Using this similarity, we can set up the following proportion:

BK / KP = BM / MN

Since BM = BK + KM, and BM = EA, we can rewrite the proportion:

EA / KP = BK + KM / MN

Since BM = EA and BK = KM, the equation becomes:

BM / KP = BK / MN

Since BM = BK + KM, we can rewrite the equation again:

BK + KM / KP = BK / MN

Simplifying further:

1 + KM / KP = 1 / MN

Since KM = KE and KE = EA, and EA = BM, we can substitute these values:

1 + EA / KP = 1 / MN

Since BM = EA:

1 + BM / KP = 1 / MN

Since MN = KP / 1 + BM / KP:

MN = KP / (1 + BM / KP)

Now, let's find the value of MN using the given lengths: BM = EA = 18 (given) KP = BC - BM = 14 - 18 = -4 (Since BM extends beyond the base BC, we need to consider the length as negative) MN = -4 / (1 + 18 / -4) MN = -4 / (1 - 4.5) MN = -4 / (-3.5) MN = 1.143

Therefore, MN = 1.143.

To find KP: Since KP || MN, we can consider triangles KPC and BKM to be similar. Using this similarity, we can set up the following proportion:

KP / KC = BK / BM

Since BK = KM and BM = EA, we can substitute these values:

KP / KC = KM / EA

Since KM = KE and KE = EA:

KP / KC = KE / EA

Since KP + KC = BC, we can rewrite the proportion:

KP / BC = KE / EA

Since BC = 14 and EA = 18:

KP / 14 = KE / 18

Since KE = EA = 18:

KP / 14 = 18 / 18

KP / 14 = 1

Multiplying both sides by 14:

KP = 14 * 1 KP = 14

Therefore, KP = 14.

To find EF: Since AD || EF, we can consider triangles ADE and FEB to be similar. Using this similarity, we can set up the following proportion:

AE / EF = ED / EB

Since AE = BM = 18 and ED = BC = 14:

18 / EF = 14 / EB

Since EB = BC + CK and BC = 14 and CK = KP = 14:

18 / EF = 14 / (14 + 14)

18 / EF = 14 / 28

Cross-multiplying:

14 * EF = 18 * 28

EF = (18 * 28) / 14

EF = 504 / 14

EF = 36

Therefore, EF = 36.

In summary: MN ≈ 1.143 KP = 14 EF = 36

Question 2

дано abcd трапеция , BM=MK=KE=EA, MN||KP||EF||AD,BC=14, AD=18. найти MN,KP,EF

Answer

To find the lengths of MN, KP, and EF in the trapezoid ABCD, we need to use the given information and the properties of trapezoids. First, let's draw a diagram to visualize the trapezoid: A ________B / \ / \ D________C Given: - BM = MK = KE = EA (the trapezoid ABCD is isosceles) - MN || KP || EF || AD (the sides MN, KP, EF, and AD are parallel) To find MN: Since MN || KP, we can consider triangles BKM and KPC to be similar. Using this similarity, we can set up the following proportion: BK / KP = BM / MN Since BM = BK + KM, and BM = EA, we can rewrite the proportion: EA / KP = BK + KM / MN Since BM = EA and BK = KM, the equation becomes: BM / KP = BK / MN Since BM = BK + KM, we can rewrite the equation again: BK + KM / KP = BK / MN Simplifying further: 1 + KM / KP = 1 / MN Since KM = KE and KE = EA, and EA = BM, we can substitute these values: 1 + EA / KP = 1 / MN Since BM = EA: 1 + BM / KP = 1 / MN Since MN = KP / 1 + BM / KP: MN = KP / (1 + BM / KP) Now, let's find the value of MN using the given lengths: BM = EA = 18 (given) KP = BC - BM = 14 - 18 = -4 (Since BM extends beyond the base BC, we need to consider the length as negative) MN = -4 / (1 + 18 / -4) MN = -4 / (1 - 4.5) MN = -4 / (-3.5) MN = 1.143 Therefore, MN = 1.143. To find KP: Since KP || MN, we can consider triangles KPC and BKM to be similar. Using this similarity, we can set up the following proportion: KP / KC = BK / BM Since BK = KM and BM = EA, we can substitute these values: KP / KC = KM / EA Since KM = KE and KE = EA: KP / KC = KE / EA Since KP + KC = BC, we can rewrite the proportion: KP / BC = KE / EA Since BC = 14 and EA = 18: KP / 14 = KE / 18 Since KE = EA = 18: KP / 14 = 18 / 18 KP / 14 = 1 Multiplying both sides by 14: KP = 14 * 1 KP = 14 Therefore, KP = 14. To find EF: Since AD || EF, we can consider triangles ADE and FEB to be similar. Using this similarity, we can set up the following proportion: AE / EF = ED / EB Since AE = BM = 18 and ED = BC = 14: 18 / EF = 14 / EB Since EB = BC + CK and BC = 14 and CK = KP = 14: 18 / EF = 14 / (14 + 14) 18 / EF = 14 / 28 Cross-multiplying: 14 * EF = 18 * 28 EF = (18 * 28) / 14 EF = 504 / 14 EF = 36 Therefore, EF = 36. In summary: MN ≈ 1.143 KP = 14 EF = 36