Question

1. which transformation maps δabc onto δdef?

a. δabc reflects across the y - axis.
b. δabc translates to the right 3 units and down 1 unit.
c. δabc rotates 90° clockwise about the origin.
d. δabc translates to the right 1 unit and down 3 units.

1. which transformation maps δdef onto δghi?

a. δdef rotates 90° counterclockwise about the origin.
b. δdef rotates 180° counterclockwise about the origin.
c. δdef reflects across the x - axis.
d. δdef reflects across the y - axis.

1. what sequence of transformations maps δabc onto δdef? select all that apply.

a. δabc rotates 180° about the origin, then translates left 2 units.
b. δabc reflects across the x - axis, then translates down 2 units.
c. δabc translates 2 units to the right, then rotates 180° about the origin.
d. δabc translates 2 units to the right, then reflects across the x - axis.

1. which sequence of transformations maps rectangle efgh onto rectangle abcd? select all that apply.

a. rectangle efgh reflects across the y - axis, then reflects across the x - axis.
b. rectangle efgh rotates 180° about the origin, then translates up 4 units.
c. rectangle efgh translates down 4 units, then rotates 180° about the origin
d. rectangle efgh translates 4 units to the right, then rotates 180° about the origin

Explanation:

Question 1

Step1: Analyze reflection over y - axis

A reflection over the y - axis changes the sign of the x - coordinate of a point \((x,y)\) to \((-x,y)\). Let's check the coordinates of \(\triangle ABC\) and \(\triangle DEF\). If we take a vertex of \(\triangle ABC\), say \(A\), \(B\), \(C\) and reflect over y - axis, the resulting points do not match the vertices of \(\triangle DEF\).

Step2: Analyze translation right 3, down 1

A translation of a point \((x,y)\) right 3 units and down 1 unit is \((x + 3,y-1)\). Let's find the coordinates of vertices of \(\triangle ABC\) and \(\triangle DEF\). Suppose \(A=(x_1,y_1)\), \(B=(x_2,y_2)\), \(C=(x_3,y_3)\) in \(\triangle ABC\) and \(D=(x_1+3,y_1 - 1)\), \(E=(x_2+3,y_2 - 1)\), \(F=(x_3+3,y_3 - 1)\) in \(\triangle DEF\) (by visually inspecting the graph), this seems to match.

Step3: Analyze 90° clockwise rotation

A 90° clockwise rotation about the origin of a point \((x,y)\) is \((y,-x)\). The shape and position after this rotation do not match \(\triangle DEF\).

Step4: Analyze translation right 1, down 3

A translation of right 1 and down 3 \((x + 1,y-3)\) does not match the coordinates of \(\triangle DEF\) vertices.

Step1: Analyze 90° counter - clockwise rotation

A 90° counter - clockwise rotation about the origin of a point \((x,y)\) is \((-y,x)\). The resulting shape and position do not match \(\triangle GHI\).

Step2: Analyze 180° counter - clockwise rotation

A 180° counter - clockwise rotation about the origin of a point \((x,y)\) is \((-x,-y)\). The vertices of \(\triangle DEF\) after 180° rotation do not match \(\triangle GHI\).

Step3: Analyze reflection over x - axis

A reflection over the x - axis changes the sign of the y - coordinate of a point \((x,y)\) to \((x,-y)\). The vertices of \(\triangle DEF\) after reflection over x - axis do not match \(\triangle GHI\).

Step4: Analyze reflection over y - axis

A reflection over the y - axis changes the sign of the x - coordinate of a point \((x,y)\) to \((-x,y)\). By visually inspecting the coordinates of vertices of \(\triangle DEF\) and \(\triangle GHI\), a reflection over the y - axis maps \(\triangle DEF\) onto \(\triangle GHI\).

Step1: Analyze option a (180° rotation then left 2)

A 180° rotation about the origin of a point \((x,y)\) is \((-x,-y)\), then translating left 2 \((-x - 2,-y)\). By checking the vertices of \(\triangle ABC\) and \(\triangle DEF\), this does not match.

Step2: Analyze option b (reflect over x - axis then down 2)

A reflection over the x - axis of a point \((x,y)\) is \((x,-y)\), then translating down 2 \((x,-y - 2)\). By visually inspecting the graph, the vertices of \(\triangle ABC\) after reflection over x - axis and translation down 2 match the vertices of \(\triangle DEF\).

Step3: Analyze option c (translate right 2 then 180° rotation)

Translating right 2 \((x + 2,y)\) then 180° rotation \((-(x + 2),-y)\) does not match the vertices of \(\triangle DEF\).

Step4: Analyze option d (translate right 2 then reflect over x - axis)

Translating right 2 \((x + 2,y)\) then reflecting over x - axis \((x + 2,-y)\) does not match the vertices of \(\triangle DEF\).

Answer:

b. \(\triangle ABC\) translates to the right 3 units and down 1 unit.

Question 2