Kathleens's ice cream cone has a diameter of 9 cm and a slant height of 7.5cm, what is the height of the ice cream cone

  6 cm

Step-by-step explanation:

The triangle formed by the slant height, the height, and the radius is a right triangle with one leg 4.5 cm and hypotenuse 7.5 cm. These dimensions are in the ratio ...

  4.5 : 7.5 = 3 : 5

so suggests that the dimensions are similar to those of a 3-4-5 right triangle with a scale factor of 1.5 cm.

That is, the height of the cone will be 1.5 cm · 4 = 6 cm.

Comment on 3-4-5 right triangle

Triangles with dimensions of these ratios often show up in algebra problems because they work out nicely, avoiding irrational numbers. It is easy enough to show that a triangle with legs 3 and 4 will have a hypotenuse of 5, according to the Pythagorean theorem:

  5^2 = 3^2 + 4^2

  25 = 9 + 16 . . . . . . true

3 is greater then both of those numbers