Triangle \(ABC\) has coordinates \(A (1, 3), B (2,0),\) and \(C (4,1).\) The image of this triangle after a sequence of transformations is triangle \(A’B’C’\) where \(A’ (\text-5,\text-3), B’ (\text-4,0),\) and \(C’ (\text-2,\text-1).\)

Write a sequence of transformations that takes triangle \(ABC\) to triangle \(A’B’C’\).

Two triangles in coordinate plane. Green triangle has point A at negative 5 comma negative 3, point B at negative 4 comma 0, point C at negative 2 comma negative 1. Blue triangle has point A at 1 comma 3, point B at 2 comma 0, point C at 4 comma 1.

Show using transformations or segment lengths that triangle \(ABC\) is congruent to triangle \(DEF\).

The density of water is 1 gram per cm³. An object floats in water if its density is less than water’s density, and it sinks if its density is greater than water’s. Will a 1.17 gram diamond in the shape of a pyramid whose base has area 2 cm² and whose height is 0.5 centimeters sink or float? Explain your reasoning. 

Technology required. An oblique cylinder with a base of radius 2 units is shown. The top of the cylinder can be obtained by translating the base by the directed line segment \(AB\) which has length 16 units. The segment \(AB\) forms a \(30^\circ\) angle with the plane of the base. What is the volume of the cylinder?

This design began from the construction of an equilateral triangle. Record at least 3 rigid transformations (rotation, reflection, translation) that take parts of the diagram to congruent parts of the diagram.

Equilateral Triangle D E F and three small congruent circles, centers A B and C. Take every triangle edge and replace its middle third with a circle. Line segment D F passes through Points A, L and K, with L and K on the circle. Line segment E F passes through Points C, I and J, with I and J on the circle. Line segment D E passes through Points B, G and H, with G and H on the circle.