Question

27)
a) reflection across the y-axis
b) translation: 1 unit right
c) rotation 90° clockwise about the origin
d) translation: 3 units right and 1 unit up
find the area of each.
28)
a) rotation 180° about the origin
b) translation: 2 units right and 6 units up
c) translation: 8 units right and 6 units up
d) reflection across the y-axis

Explanation:

Problem 27:

Step1: Analyze each transformation option

• Option A (Reflection across y - axis): A reflection across the y - axis changes the sign of the x - coordinate of a point \((x,y)\) to \((-x,y)\). By visually inspecting the graph, the positions of the vertices do not match the pattern of a y - axis reflection.
• Option B (Translation: 1 unit right): A translation of 1 unit right would move each point \((x,y)\) to \((x + 1,y)\). The relative positions of the figures do not suggest a 1 - unit right translation.
• Option C (Rotation \(90^{\circ}\) clockwise about the origin): A \(90^{\circ}\) clockwise rotation about the origin transforms a point \((x,y)\) to \((y,-x)\). The shape and position of the figures do not match the result of a \(90^{\circ}\) clockwise rotation.
• Option D (Translation: 3 units right and 1 unit up): By observing the corresponding vertices of the two figures, if we take a vertex of the original figure and move it 3 units to the right (increase x - coordinate by 3) and 1 unit up (increase y - coordinate by 1), we get the position of the corresponding vertex in the transformed figure. This matches the visual shift of the figure.

Step1: Analyze each transformation option

• Option A (Rotation \(180^{\circ}\) about the origin): A \(180^{\circ}\) rotation about the origin transforms a point \((x,y)\) to \((-x,-y)\). The orientation and position of the triangles do not match the result of a \(180^{\circ}\) rotation.
• Option B (Translation: 2 units right and 6 units up): A translation of 2 units right and 6 units up would move each point \((x,y)\) to \((x + 2,y + 6)\). The distance between corresponding vertices is not consistent with this translation.
• Option C (Translation: 8 units right and 6 units up): By looking at the horizontal and vertical distances between corresponding vertices (e.g., vertex \(H\) and \(H'\), \(G\) and \(G'\), \(F\) and \(F'\)), the horizontal distance is 8 units (from left - side x - coordinate to right - side x - coordinate) and vertical distance is 6 units (from lower y - coordinate to upper y - coordinate). This matches the translation of 8 units right and 6 units up.
• Option D (Reflection across the y - axis): A y - axis reflection would change the sign of the x - coordinate, but the orientation of the triangle (the "direction" it is facing) does not match a y - axis reflection.

Answer:

D) translation: 3 units right and 1 unit up

Problem 28: