02.07 Review and Practice Exam for Geometry

Pentagon VWXYZ is shown on the coordinate plane below. If pentagon VWXYZ is rotated 270° clockwise around the origin, what are the coordinates of the endpoints of the side congruent to side YZ in the image pentagon?

The side of the image congruent to side YZ in the pre-image will be side Y'Z'. Recall that rotating a figure 270° clockwise is the same as rotating a figure 90° counterclockwise. Therefore, you will apply the rotation rule (x, y) → (−y, x) to points Y and Z.

Pentagon ZABCD is shown on the coordinate plane. If pentagon ZABCD is reflected across the x-axis to create pentagon Z'A'B'C'D', what is the ordered pair of point Z'?

There are two ways to find the ordered pair of point Z'. First, you could remember the rule that points reflected across the x-axis will have identical x-coordinates and the sign on the y-coordinate will change. In other words, (x, y) → (x, −y). Since point Z is at (7, 3), its reflected point will be (7, −3). Or, you could count the distance between point Z and the x-axis. The reflected point Z' must be the same distance away in the same direction.

Trapezoids PQRS and P'Q'R'S' are shown on the coordinate plane. What rotation was applied to trapezoid PQRS to create trapezoid P'Q'R'S'?

The points of trapezoids PQRS and P'Q'R'S' are at the following ordered pairs: PQRS (1, 1) (2, 2) (4, 2) (5, 1) P'Q'R'S' (−1, −1) (−2, −2) (−4, −2) (−5, −1) The opposites of each x- and y-coordinate in the pre-image are used to create the image. In mathematical terms, (x, y) → (−x, −y). Therefore, trapezoid PQRS has been rotated 180° to create trapezoid P'Q'R'S'. Y(5, 0) → Y'(0, 5) Z(3, −1) → Z'(1, 3) The coordinates of the endpoints of the side congruent to side YZ in the image pentagon are (0, 5) and (1, 3). Take a look at the image.

Triangle ABC and its transformation, triangle A'B'C', are shown on the coordinate plane. Which two transformations are applied to triangle ABC to create A'B'C'?

Because point A is not on the bottom of the triangle, triangle ABC was not reflected across the x-axis. Since triangle ABC passes through the line y = x, it could not have been reflected across this line. If it had passed through the line, point A' would be in the bottom right corner. Therefore, triangle ABC must have been reflected across the y-axis. To reach triangle A'B'C' from the reflected image of triangle A1B1C1, it must be translated. Count the number of units and the direction between one of the points in triangle A1B1C1 to the translated point in triangle A'B'C'. Let's use point B1 and B'. Because you must count left 1 place from point B1 to point B', the rule for the x-coordinate is x − 1. Because you are counting up 6 places, the rule for the y-coordinate is y + 6. The translation rule is (x, y) → (x − 1, y + 6). Triangle ABC is reflected across the y-axis and translated according to the rule (x, y) → (x − 1, y + 6) to reach triangle A'B'C'.

Parallelogram FGHI and its reflection, parallelogram F'G'H'I', are shown below in the coordinate plane. What is the line of reflection between parallelograms FGHI and F'G'H'I'?

Because the parallelograms are somewhat side-by-side, the line of reflection will be a vertical line. Notice the two parallelograms are not the same distance away from the y-axis. Therefore, this is not the line of reflection. Find the distance between points G and G' and H and H'. Points G and G' are 8 units apart while points H and H' are 10 units apart. Divide each of these values by 2 to achieve 4 and 5 units respectively. So count 4 units from point G and G' and count 5 units from point H and H'. All points of parallelogram FGHI and parallelogram F'G'H'I' are equidistant from x = 1; therefore, this is the line of reflection.

Hexagon NOPQRS is translated on the coordinate plane to create hexagon N'O'P'Q'R'S'. If kite JKLM is translated according to the same rule that translated hexagon NOPQRS, what is the ordered pair of point K'?

Before you can find the ordered pair of K', you need to find the rule that translates hexagon NOPQRS into hexagon N'O'P'Q'R'S'. Choose any point on hexagon NOPQRS. Let's choose point P. Count the places it takes to get from point P to point P'. Count left/right first and then up/down. Because you must count right 4 units and up 2 units from point P to point P', the rule for the x-coordinate is x + 4, and the rule for the y-coordinate is y + 2. Therefore, the translation rule between hexagons NOPQRS and N'O'P'Q'R'S' is (x, y) → (x + 4, y + 2). Now, apply this rule to point K. (x, y) → (x + 4, y + 2) (−1, 5) → (−1 + 4, 5 + 2) (−1, 5) → (3, 7) Point K' is at (3, 7). Take a look at the entire translated kite J'K'L'M'.

Rhombus EFGH is shown on the coordinate plane. If rhombus EFGH is reflected across the y-axis to create rhombus E'F'G'H', what is the ordered pair of point G'?

First, you could remember the rule that points reflected across the y-axis will have the opposite sign on the x-coordinate and identical y-coordinates. In other words, (x, y) → (−x, y). Since point G is at (5, −5), its reflected point will be (−5, −5).

Side-Side-Side Postulate (SSS)

If the sides of one triangle are congruent to the sides of a second triangle, then the triangles are congruent.

Angle-Side-Angle (ASA) Postulate

If two angles and the included side of one triangle are congruent to two angles and the included side of another triangle, the triangles are congruent.

Hypotenuse-Leg (HL) Theorem

If two right triangles have a congruent hypotenuse and a pair of congruent, corresponding sides, then the two triangles are congruent.

Side-Angle-Side (SAS) Postulate

If two sides and the included angle of one triangle are congruent to two sides and the included angle of another triangle, then the triangles are congruent.

Triangle XYZ is shown on the coordinate plane. If triangle XYZ is translated according to the rule (x, y) → (x + 3, y + 9), what are the coordinates of point Z'? There are two ways of finding the coordinates of point Z'.

METHOD ONE One way is to apply the translation rule to the coordinates of point Z, (4, −5). (x, y) → (x + 3, y + 9) (4, −5) → (4 + 3, −5 + 9) (4, −5) → (7, 4) The coordinates of point Z' are (7, 4).

Rectangle TUVW is translated on the coordinate plane to create rectangle T'U'V'W'. What is the rule representing the translation of rectangle TUVW to rectangle T'U'V'W'?

To find the translation rule between the two rectangles, select any point on rectangle TUVW. Let's pick point V. Then count the places it takes to get from point V to point V,' and note the direction you count. Count left/right first and then up/down. Because you must count left 8 places from point V to point V', the rule for the x-coordinate is x − 8. Because you are counting down 7 places, the rule for the y-coordinate is y − 7. Therefore, the translation rule between rectangles TUVW and T'U'V'W' is (x, y) → (x − 8, y − 7).

Heptagon ABCDEFG is shown on the coordinate plane below. If heptagon ABCDEFG is rotated 90° clockwise around the origin to create heptagon A'B'C'D'E'F'G', what is the ordered pair of point G'?

To rotate a point 90° clockwise, follow the rotation rule of (x, y) → (y, −x). Reverse the x- and y-coordinates, and use the opposite sign of the x-coordinate. Point G has an ordered pair of (3, 1). Therefore, point G' is at the ordered pair of (1, −3).

Trapezoid IJKL is shown on the coordinate plane. If trapezoid IJKL is reflected across the line y = x to create trapezoid I'J'K'L', what are the vertices of trapezoid I'J'K'L'?

When a figure is reflected across the line y = x, the x and y values are reversed. In other words, (x, y) → (y, x). VERTEX I (x, y) → (y, x) (3, 4) → (4, 3) I': (4, 3) VERTEX J (x, y) → (y, x) (6, 4) → (4, 6) J': (4, 6) VERTEX K (x, y) → (y, x) (7, 2) → (2, 7) K': (2, 7) VERTEX L (x, y) → (y, x) (2, 2) → (2, 2) L': (2, 2)