Geometric Mean and Altitude in Triangles

The video tutorial explains the concept of the geometric mean in right triangles, highlighting its role as the altitude that divides a right triangle into two smaller similar triangles. It demonstrates how the geometric mean is used in proportions to determine the lengths of triangle sides. The tutorial includes examples and applications in architecture, particularly in designing trusses. It also covers specific cases like isosceles right triangles, providing a comprehensive understanding of the geometric mean's significance in geometry.

What is the geometric mean in the context of right triangles?

• The angle opposite the hypotenuse

• The longest side of the triangle

• The altitude of the triangle

• The sum of all sides

How does the altitude affect the division of a right triangle?

• It creates two smaller triangles that are not similar

• It divides the triangle into two smaller similar triangles

• It doubles the size of the original triangle

• It changes the angles of the original triangle

If the altitude of a right triangle is 8, what is the geometric mean of the other sides?

• 10

• 16

• 8

• 4

In a right triangle, if one horizontal side is 5 and the other is 20, what is the altitude?

• 15

• 25

• 10

• 20

Why is the geometric mean important for architects and engineers?

• It helps in calculating the area of a triangle

• It determines the color of the building

• It controls the height, length, and angles of structures

• It is used to design circular structures

What is the altitude of an isosceles right triangle if the hypotenuse is 50 cm?

• 30 cm

• 25 cm

• 20 cm

• 35 cm

How does the altitude of an isosceles right triangle relate to its hypotenuse?

• It is twice the hypotenuse

• It is equal to the hypotenuse

• It is half of the hypotenuse

• It divides the hypotenuse into two equal parts

If the altitude of triangle ABC is 45 and side CD is 75, what is the length of the hypotenuse?

• 102

• 100

• 105

• 110

What is the result of multiplying the geometric mean by itself in the context of right triangles?

• The product of the two horizontal sides

• The perimeter of the triangle

• The area of the triangle

• The square of the hypotenuse

What does the geometric mean define in a right triangle?

• The angle measures

• The type of triangle

• The color of the triangle

• The proportion of the triangle's sides