Understanding Medians and Centroids in Triangles

The video tutorial explores the concept of medians in a triangle, explaining how they are drawn from a vertex to the midpoint of the opposite side. It highlights the property that all medians intersect at a single point called the centroid. The tutorial further demonstrates that these medians divide the triangle into six smaller triangles of equal area, providing a geometric proof for this property.

What is a median in a triangle?

• A line segment from a vertex to the midpoint of the opposite side

• A line segment that bisects an angle

• A line segment that is perpendicular to the opposite side

• A line segment from a vertex to the opposite vertex

What is the point called where all medians of a triangle intersect?

• Incenter

• Centroid

• Circumcenter

• Orthocenter

What unique property does the centroid of a triangle have in physics?

• It is the point where all angles are equal

• It is the point where the triangle balances

• It is the point where the triangle's perimeter is minimized

• It is the point where the triangle's area is maximized

How many smaller triangles are formed when medians divide a triangle?

• Five

• Six

• Four

• Three

What is the relationship between the areas of the six smaller triangles formed by the medians?

• They form a geometric progression

• They form an arithmetic progression

• They all have different areas

• They all have the same area

Which principle is used to prove that the smaller triangles have equal areas?

• Area = base * height

• Law of Sines

• Pythagorean Theorem

• Law of Cosines

In an obtuse triangle, where does the altitude from a vertex lie?

• Inside the triangle

• Outside the triangle

• On the hypotenuse

• On the base

What is the result when you subtract y from both sides of the equation 2z + y = 2x + y?

• 2z = 2x

• z = x

• y = x

• z = y

What is the final conclusion about the areas of the smaller triangles?

• They are proportional to the angles

• They are proportional to the sides

• They are all equal

• They are all different

What is the significance of the centroid in dividing the triangle?

• It divides the triangle into three equal parts

• It divides the triangle into six equal area triangles

• It divides the triangle into four equal area triangles

• It divides the triangle into two equal parts