Geometry Competency Assessment

A triangle has vertices A(1, 2), B(3, 6), and C(5, 2). Describe a sequence of rigid motions that maps this triangle onto a congruent triangle located in the third quadrant. Explain how the rigid motions preserve congruency.

Which of the following justifies the conclusion that two triangles are congruent based on the information provided?

Given quadrilateral PQRS with vertices at P(1, 2), Q(4, 6), R(7, 2), and S(4, -2), determine whether PQRS is a rhombus. Justify your answer using coordinate geometry.

Which transformation or sequence of transformations always produces a figure similar to the original?

Given a circle with center O and a radius of 5 units, a chord AB is 6 units long. Calculate the perpendicular distance from O to chord AB.

A trapezoid has bases of 10 cm and 14 cm, and a height of 6 cm. What is the area of the trapezoid?

A ladder leans against a vertical wall, forming a 65° angle with the ground. If the foot of the ladder is 6 feet from the wall, how tall is the ladder? Round your answer to the nearest tenth.

Which right triangle lengths show incorrect values for a, b, and c according to the Pythagorean Theorem?

Solve for x. Explain how you would do each step.