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?

• Two pairs of angles and one pair of non-included sides are congruent

• Two pairs of sides and one pair of included angles are congruent

• Three pairs of sides are congruent

• Two pairs of sides and one non-included angle are congruent

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?

• Translation

• Reflection

• Dilation

• Rotation

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?

• 72 cm²

• 144 cm²

• 48 cm²

• 84 cm²

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.

• Four times total

• Fives times total

• Two times total

• Six times total

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

• Triangle A:

  a = 8 , b = 15, c=17

• Triangle B:

  a = 3, b = 4, c = 5

• Triangle C:

  a = 8, b = 9, c = 10

• Triangle D:

  a = 6, b = 8, c = 10

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