Question

What could be the ordered pairs representing vertices C and D of quadrilateral abcd so that the new fire department is congruent to the old fire department

Answer 1

Given that a city grid of Anytown, USA is shown on the grid below. The fire department is represented by quadrilateral RSTU. Another fire department is opening in a different part of the city to maximize fire protection. The size of the new department's property must be congruent to the older department. Vertices A and B are plotted on the grid to represent two vertices of the new fire department quadrilateral ABCD.

Map of Anytown; the line y equals 7 is Main Road; the line y equals 3 is Rose Lane; the line y equals negative 1 is Crystal Avenue; the line x equals negative 4 is Brick Street; the diagonal line is Dogwood Drive; Old Fire Department forms quadrilateral with ordered pairs R at negative 4, 7, S at negative 1, 7; T at negative 1, 3; U at negative 4, 3; A is at 4, negative 1 and B is at 1, negative 1

Recall that two shapes are said to be congruent if the lengths of the sides and the angles are the same.

For quadrilateral RSTU to be congruent to quadrilateral ABCD, then RS = AB, RU = AD, ST = BC and TU = CD

Also, the figure ABCD is a result of a rigid body translation of the figure RSTU.

Given that R is at point (-4, 7) and A is at point (4, -1), also given that S is at point (-1, 7) and B is at point (1, -1).

It can be seen that points AB is a result of refrecting points RS across the y-axis and then shifting the resulting points down by 8 units.

Thus, given point T as (-1, 3) and point U as (-4, 3), refrection of points T and U across the y-axis will result in points (1, 3) and (4, 3), then shifting the resulting points down by 8 units will result to points (1, -5) and (4, -5)

Therefore, the ordered pairs representing vertices C and D of quadrilateral ABCD so that the new fire department is congruent to the old fire department are C(1, -5) and D(4, -5).