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ABC is a triangle. D and E are the mid points of AB and BC and P is any point on AC, if M and N are the mid points of AP and PC, then find DM : EN.
- a)3 : 1
- b)1 : 1
- c)1 : 2
- d)2 : 1
Correct answer is option 'B'. Can you explain this answer?
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To find the ratio DM : EN, we need to consider the properties of midpoints and use them to determine the lengths of DM and EN.
Properties of Midpoints:
1. The line segment joining the midpoints of two sides of a triangle is parallel to the third side and half its length.
2. The line segment joining the midpoints of two sides of a triangle divides the third side into two equal parts.
Using these properties, we can determine the lengths of DM and EN.
Solution:
Let's assume that AB = BC = AC = 2 units (for simplicity).
1. Finding the lengths of DM and EN:
- Since D is the midpoint of AB, DM = AD.
- Similarly, since E is the midpoint of BC, EN = CE.
- Since AB = BC = AC = 2 units, we can determine the lengths of AD and CE using the property mentioned above.
- AD = DM = AB/2 = 2/2 = 1 unit.
- CE = EN = BC/2 = 2/2 = 1 unit.
2. Finding the lengths of AP and PC:
- Since M is the midpoint of AP, AM = MP.
- Similarly, since N is the midpoint of PC, CN = NP.
- AP = AC - PC = 2 - PC.
- PC = AC - AP = 2 - AP.
- Using the property mentioned above, we can determine the lengths of AM and CN.
- AM = MP = AP/2 = (2 - PC)/2 = (2 - (2 - AP))/2 = AP/2 = 1/2 * AP.
- CN = NP = PC/2 = (2 - AP)/2 = 1 - AP/2.
3. Finding the lengths of DM and EN in terms of AP:
- Since DM = AD = 1 unit and AM = 1/2 * AP, we can write DM in terms of AP as DM = 1 + 1/2 * AP.
- Similarly, since EN = CE = 1 unit and CN = 1 - AP/2, we can write EN in terms of AP as EN = 1 + (1 - AP/2) = 2 - AP/2 = 2 - 1/2 * AP.
4. Simplifying the ratios DM : EN:
- DM = 1 + 1/2 * AP.
- EN = 2 - 1/2 * AP.
- Dividing both sides by AP, we get DM/AP : EN/AP = (1 + 1/2 * AP)/(2 - 1/2 * AP).
- Simplifying further, we get DM/AP : EN/AP = (2 + AP)/(4 - AP).
Since the ratio DM : EN is in terms of AP, we cannot determine the exact ratio without knowing the value of AP. However, we can see that the ratio DM : EN is not equal to 1 : 1, 1 : 2, or 2 : 1. Therefore, the correct answer is option 'B' - 1 : 1.
Properties of Midpoints:
1. The line segment joining the midpoints of two sides of a triangle is parallel to the third side and half its length.
2. The line segment joining the midpoints of two sides of a triangle divides the third side into two equal parts.
Using these properties, we can determine the lengths of DM and EN.
Solution:
Let's assume that AB = BC = AC = 2 units (for simplicity).
1. Finding the lengths of DM and EN:
- Since D is the midpoint of AB, DM = AD.
- Similarly, since E is the midpoint of BC, EN = CE.
- Since AB = BC = AC = 2 units, we can determine the lengths of AD and CE using the property mentioned above.
- AD = DM = AB/2 = 2/2 = 1 unit.
- CE = EN = BC/2 = 2/2 = 1 unit.
2. Finding the lengths of AP and PC:
- Since M is the midpoint of AP, AM = MP.
- Similarly, since N is the midpoint of PC, CN = NP.
- AP = AC - PC = 2 - PC.
- PC = AC - AP = 2 - AP.
- Using the property mentioned above, we can determine the lengths of AM and CN.
- AM = MP = AP/2 = (2 - PC)/2 = (2 - (2 - AP))/2 = AP/2 = 1/2 * AP.
- CN = NP = PC/2 = (2 - AP)/2 = 1 - AP/2.
3. Finding the lengths of DM and EN in terms of AP:
- Since DM = AD = 1 unit and AM = 1/2 * AP, we can write DM in terms of AP as DM = 1 + 1/2 * AP.
- Similarly, since EN = CE = 1 unit and CN = 1 - AP/2, we can write EN in terms of AP as EN = 1 + (1 - AP/2) = 2 - AP/2 = 2 - 1/2 * AP.
4. Simplifying the ratios DM : EN:
- DM = 1 + 1/2 * AP.
- EN = 2 - 1/2 * AP.
- Dividing both sides by AP, we get DM/AP : EN/AP = (1 + 1/2 * AP)/(2 - 1/2 * AP).
- Simplifying further, we get DM/AP : EN/AP = (2 + AP)/(4 - AP).
Since the ratio DM : EN is in terms of AP, we cannot determine the exact ratio without knowing the value of AP. However, we can see that the ratio DM : EN is not equal to 1 : 1, 1 : 2, or 2 : 1. Therefore, the correct answer is option 'B' - 1 : 1.
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