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Let's go through each question and solve them.

### 1) **The medians of the triangle intersect at _______**
The medians of a triangle intersect at a single point called the **centroid**. The centroid is the point where the three medians meet, and it divides each median into two segments, with the longer one being twice the length of the shorter one.

**Answer:** Centroid

### 2) **The point of intersection of the medians of the triangle divides each of them by the ratio _______ : _______ from the vertex.**
The centroid divides each median in a **2:1** ratio. The part of the median from the vertex to the centroid is twice as long as the part from the centroid to the midpoint of the opposite side.

**Answer:** 2:1

### 3) **If \( \overline{AD} \) is a median in \( \triangle ABC \) and M is the point of intersection of its medians, \( AM = 6 \, \text{cm} \), then \( AD = \dots \, \text{cm} \).**
Since \( AM \) is the distance from the vertex \( A \) to the centroid \( M \), and the centroid divides the median in a ratio of 2:1, the total length of the median \( AD \) is three times \( AM \). Thus:
\[
AD = 3 \times AM = 3 \times 6 \, \text{cm} = 18 \, \text{cm}.
\]

**Answer:** \( AD = 18 \, \text{cm} \).

### 4) **The length of the hypotenuse in a thirty and sixty triangle equals _______ the length of the side opposite the angle whose measure is \( 30^\circ \).**
In a 30°-60°-90° triangle, the side opposite the 30° angle is half the length of the hypotenuse. Therefore, the hypotenuse is twice the length of the side opposite the 30° angle.

**Answer:** 2 times

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Let me know if you need more details or have questions!

Here are 5 related questions to deepen your understanding:
1. What is the formula for finding the centroid of a triangle in coordinate geometry?
2. How can we prove that the medians of a triangle intersect at a single point (the centroid)?
3. In a 30°-60°-90° triangle, what is the relationship between the side opposite the 60° angle and the hypotenuse?
4. Can you calculate the medians' intersection point for an isosceles or equilateral triangle?
5. What is the centroid's significance in physical applications, such as in balancing objects?

**Tip:** In a 30°-60°-90° triangle, remember that the side lengths always follow the ratio \( 1 : \sqrt{3} : 2 \).

1) The medians of the triangle intersect at _______. 2) The point of intersection of the medians of the triangle divides each of them by the ratio _______ : _______ from the vertex. 3) If AD is a median in triangle ABC and M is the point of intersection of its medians, AM = 6 cm. , then AD = _______ cm. 4) The length of the hypotenuse in a thirty and sixty triangle equals _______ the length of the side opposite the angle whose measure is 30°.

Learn how to find the point of intersection of medians in a triangle, understand how the centroid divides the median in a 2:1 ratio, and solve for the hypotenuse in a 30°-60°-90° triangle. Suitable for Grades 9-10 students.

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