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### 1) In \( \triangle ABC \), if \( AD \) is a median and \( M \) is the point of intersection of its medians and \( DM = 4 \, \text{cm} \), then \( AD = \) ?

**Given**:
- \( AD \) is a median (so it divides the opposite side \( BC \) into two equal parts).
- \( M \) is the centroid of the triangle, and it divides the median in the ratio 2:1.

**Solution**:
Since \( M \) is the centroid, \( DM = \frac{1}{3}AD \). Therefore, \( AD = 3 \times DM = 3 \times 4 \, \text{cm} = 12 \, \text{cm} \).

**Answer**: \( (b) \, 12 \, \text{cm} \)

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### 2) In \( \triangle ABC \) where \( \angle BAC = 90^\circ \) and \( BC = 12 \, \text{cm} \), \( AD \) is a median and \( M \) is the point of intersection of its medians. Find \( AM \).

**Given**:
- \( \triangle ABC \) is a right triangle with \( \angle BAC = 90^\circ \) and \( BC = 12 \, \text{cm} \).
- \( AD \) is a median, meaning \( D \) is the midpoint of \( BC \), so \( BD = DC = 6 \, \text{cm} \).

**Solution**:
In a right triangle, the median from the right angle vertex to the hypotenuse is half the hypotenuse. So, \( AM = \frac{1}{2} \times 12 \, \text{cm} = 6 \, \text{cm} \).

**Answer**: \( (d) \, 6 \, \text{cm} \)

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### 3) The length of the median drawn from the vertex of the right angle in the right-angled triangle is \( \dots \) the length of the hypotenuse of the triangle.

**Solution**:
In any right triangle, the median drawn from the right angle vertex to the hypotenuse is half the length of the hypotenuse.

**Answer**: \( (b) \, \frac{1}{2} \)

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### 4) In \( \triangle ABC \), if \( AD \) is a median and \( M \) is the point of concurrence of the medians, then \( AM = \dots AD \).

**Given**:
- \( AD \) is a median.
- \( M \) is the centroid, which divides \( AD \) in the ratio 2:1, where \( AM = \frac{2}{3}AD \).

**Solution**:
Using the property of the centroid, \( AM = \frac{2}{3} \times AD \).

**Answer**: \( (c) \, \frac{2}{3} \)

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Let me know if you need any details or further clarifications on any of the questions. 

Here are some follow-up questions that might help deepen your understanding:

1. How can you find the centroid in an isosceles triangle?
2. What is the relationship between the median and the area of the triangle?
3. Why does the centroid divide the median into a 2:1 ratio?
4. How do you calculate the area of a triangle using medians?
5. In a scalene triangle, can the centroid and the orthocenter coincide?

**Tip**: Remember, the centroid is always located two-thirds of the way from a vertex along the median, dividing the median in a 2:1 ratio.

In ∆ABC, several geometric properties involving medians and centroids are explored through four different sub-questions about triangle properties, medians, centroids, and right-angled triangles.

This problem explores the relationship between medians and centroids in a triangle. It covers geometric properties of medians, centroids, and their application in right-angled triangles, including key theorems and formulas. Suitable for Grades 9-10 students studying geometry.

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