Question

Дано: AM=MB, CN=ND, MN - 21см MK:KP:PN = 2:3:2 Найти AD и BC

Answer 1

2х+3х+2х=7х
21см / х = 7х 
21/7
х=3

Answer 2

Given Information:

We are given the following information: - AM = MB - CN = ND - MN = -21 cm - MK:KP:PN = 2:3:2

We need to find the values of AD and BC.

Approach:

To solve this problem, we can use the concept of similar triangles. By examining the given information, we can see that triangles AMK and BKP are similar, as well as triangles CND and PNB. This allows us to set up proportions and solve for the unknown values.

Solution:

Let's start by finding the value of AD.

Since triangles AMK and BKP are similar, we can set up the following proportion:

AM / BK = MK / KP

Substituting the given values: AM / BK = -21 / 3

Since AM = MB, we can substitute AM with MB: MB / BK = -21 / 3

Simplifying the equation: MB / BK = -7

Since MK:KP:PN = 2:3:2, we can express BK in terms of KP: BK = KP * (2/3)

Substituting this value into the equation: MB / (KP * (2/3)) = -7

Simplifying further: MB / KP = -7 * (3/2)

Now, let's find the value of BC.

Since triangles CND and PNB are similar, we can set up the following proportion:

CN / PB = ND / PN

Substituting the given values: CN / PB = -21 / 2

Since CN = ND, we can substitute CN with ND: ND / PB = -21 / 2

Simplifying the equation: ND / PB = -10.5

Since MK:KP:PN = 2:3:2, we can express PB in terms of KP: PB = KP * (3/2)

Substituting this value into the equation: ND / (KP * (3/2)) = -10.5

Simplifying further: ND / KP = -10.5 * (2/3)

Now we have two equations: 1. MB / KP = -7 * (3/2) 2. ND / KP = -10.5 * (2/3)

Since MB = AM and ND = CN, we can rewrite the equations as: 1. AM / KP = -7 * (3/2) 2. CN / KP = -10.5 * (2/3)

Now, let's solve these equations simultaneously to find the value of KP.

By multiplying equation 1 by 2 and equation 2 by 3, we can eliminate KP: 1. 2 * (AM / KP) = -7 * 3 2. 3 * (CN / KP) = -10.5 * 2

Simplifying further: 1. (2AM) / KP = -21 2. (3CN) / KP = -21

Since AM = MB and CN = ND, we can rewrite the equations as: 1. (2MB) / KP = -21 2. (3ND) / KP = -21

Now, let's add these two equations together: (2MB + 3ND) / KP = -21 + (-21)

Simplifying further: (2MB + 3ND) / KP = -42

Since AM = MB and CN = ND, we can rewrite the equation as: (2AM + 3CN) / KP = -42

Now, let's substitute the given values: (2 * (-21) + 3 * (-21)) / KP = -42

Simplifying further: (-42 + (-63)) / KP = -42

Combining like terms: (-105) / KP = -42

Now, let's solve for KP: KP = (-105) / (-42) = 2.5

Now that we have the value of KP, we can substitute it back into the equations to find the values of AD and BC.

Substituting KP = 2.5 into equation 1: AM / 2.5 = -7 * (3/2)

Simplifying further: AM / 2.5 = -10.5

Multiplying both sides by 2.5: AM = -10.5 * 2.5 = -26.25

Since AM = MB, we can conclude that: MB = -26.25

Substituting KP = 2.5 into equation 2: CN / 2.5 = -10.5 * (2/3)

Simplifying further: CN / 2.5 = -7

Multiplying both sides by 2.5: CN = -7 * 2.5 = -17.5

Since CN = ND, we can conclude that: ND = -17.5

Therefore, the values of AD and BC are: AD = MB = -26.25 BC = ND = -17.5

Please note that the negative sign indicates the direction of the line segment, but the magnitude represents the length.

I hope this helps! Let me know if you have any further questions.

Answer 3

Problem Analysis

We are given the following information: - AM = MB - CN = ND - MN = -21 cm - MK:KP:PN = 2:3:2

We need to find the values of AD and BC.

Solution

To find the values of AD and BC, let's analyze the given information step by step.

1. AM = MB: This means that the line segment AM is equal in length to the line segment MB. Since the lengths are equal, we can consider AM and MB as two equal parts of a whole line segment. Let's call the length of AM and MB as x.

2. CN = ND: Similar to the previous step, this means that the line segment CN is equal in length to the line segment ND. Let's call the length of CN and ND as y.

3. MN = -21 cm: This means that the line segment MN has a length of -21 cm. Since lengths cannot be negative, we can assume that the direction of MN is opposite to the direction of the other line segments. Let's call the length of MN as z.

4. MK:KP:PN = 2:3:2: This ratio tells us the relative lengths of the line segments MK, KP, and PN. Let's call the common ratio between these line segments as r. Therefore, we have MK = 2r, KP = 3r, and PN = 2r.

Now, let's use these relationships to find the values of AD and BC.

5. AD: To find the value of AD, we need to find the sum of the lengths of the line segments AM, MN, and ND. From step 1, we know that AM = MB = x. From step 3, we know that MN = -21 cm = z. From step 2, we know that CN = ND = y. Therefore, AD = AM + MN + ND = x + z + y.

6. BC: To find the value of BC, we need to find the sum of the lengths of the line segments CN, NK, and KP. From step 2, we know that CN = ND = y. From step 4, we know that NK = 2r. From step 4, we know that KP = 3r. Therefore, BC = CN + NK + KP = y + 2r + 3r.

Now, we have expressions for AD and BC in terms of x, y, z, and r. We can substitute the given values and solve for AD and BC.

Let's summarize the steps:

1. AM = MB = x 2. CN = ND = y 3. MN = -21 cm = z 4. MK:KP:PN = 2:3:2, which implies MK = 2r, KP = 3r, and PN = 2r 5. AD = AM + MN + ND = x + z + y 6. BC = CN + NK + KP = y + 2r + 3r

Now, we can substitute the given values and solve for AD and BC.

Calculation

Given that MN = -21 cm, we can assume that z = 21 cm (since lengths cannot be negative).

Let's assume a value for r. For simplicity, let's assume r = 1.

Substituting the values into the expressions for AD and BC:

AD = x + z + y = x + 21 cm + y

BC = y + 2r