https://file.aiquickdraw.com/mathbot/file/1728572346986ifap65s9.jpg

Let's go through the four questions in the image you provided:

1. **The medians of the triangle intersect at...**  
   - The medians of a triangle intersect at the **centroid**.  
     **Answer:** The 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 the ratio **2:1**, where the longer segment is closer to the vertex.  
     **Answer:** 2:1.

3. **If \( AD \) is a median in \( \triangle ABC \) and \( M \) is the point of intersection of its medians, \( AM = 6 \text{ cm} \), then \( AD = ... \text{cm}.**  
   - Since the centroid divides the median in the ratio 2:1, \( AM = \frac{2}{3} AD \). To find \( AD \), we use:
     \[
     AD = \frac{AM \times 3}{2} = \frac{6 \times 3}{2} = 9 \text{ cm}
     \]
     **Answer:** 9 cm.

4. **The length of the hypotenuse in a 30-60-90 triangle equals ... the length of the side opposite the angle whose measure is \( 30^\circ \).**  
   - In a 30-60-90 triangle, the hypotenuse is **twice** the length of the side opposite the \( 30^\circ \) angle.  
     **Answer:** Twice.

Let me know if you want further details or have any specific questions!

### Related Questions:
1. How can we prove that the centroid divides the median in a 2:1 ratio?
2. Can we use a similar method to find the length of other triangle elements from the centroid?
3. What is the significance of the centroid in terms of balance or center of mass?
4. How do you derive the side ratios in a 30-60-90 triangle?
5. How would you calculate the other two sides of a 30-60-90 triangle if only one side length is given?

### Tip:
In any triangle, the centroid is the point where the triangle's medians meet, and it's always located inside the triangle.

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 30-60 triangle equals _______ the length of the side opposite the angle whose measure is 30 degrees.

Explore the properties of triangle medians and special right triangles. Learn how the centroid divides the median in a 2:1 ratio, and how the hypotenuse in a 30-60-90 triangle is twice the length of the side opposite the 30-degree angle. Suitable for students in Grades 8-10.

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