Question

drag the image to the correct location.
triangle abc is rotated 90° counterclockwise about point p to create triangle def. determine the correct orientation and location of triangle def.

Explanation:

Step1: Recall rotation rules

A 90° counterclockwise rotation about a point \((x,y)\) transforms a point \((a,b)\) to \((-b + x + y, a - x + y)\) (simplified: for rotation about origin, \((x,y)\to(-y,x)\); adjust for point \(P\)). We analyze the orientation of \(\triangle ABC\) (with vertices \(A,B,C\)) and how rotating 90° counterclockwise about \(P\) affects its shape.

Step2: Analyze triangle orientation

• Original \(\triangle ABC\): Let's consider the order of vertices (e.g., \(A\) at bottom, \(B\) at top, \(C\) at right).
• After 90° counterclockwise rotation about \(P\), the horizontal/vertical directions flip. The correct \(\triangle DEF\) should have a vertex arrangement matching the rotation: the "point" of the triangle (like \(C\) in \(ABC\)) should rotate such that the sides align with 90° rotation.

Looking at the options, the middle right triangle (with \(F\) on left, \(E\) on top, \(D\) on bottom - wait, no, re - evaluate: when rotating 90° counterclockwise, a horizontal side (e.g., \(AB\)) becomes vertical, and vertical side becomes horizontal (reverse direction). The correct triangle is the one where the orientation matches the 90° counterclockwise rotation of \(ABC\) about \(P\). The middle right triangle (second from top on right, with \(F\), \(E\), \(D\) arranged such that the rotation from \(ABC\) (with \(A\) below \(B\), \(C\) to the right of \(B\)) would rotate \(B\) to \(E\), \(A\) to \(D\), \(C\) to \(F\) (or similar vertex mapping) with the correct angle.

Answer:

The correct triangle is the middle - right one (the second triangle from the top on the right - hand side, with vertices \(F\), \(E\), \(D\) where \(F\) is on the left, \(E\) on the top, and \(D\) on the bottom - right, forming a triangle with the same shape as \(ABC\) but rotated 90° counterclockwise about \(P\)). If we consider the three right - hand triangles:

1. Top right: \(E\) and \(D\) horizontal, \(F\) below - not matching 90° counterclockwise of \(ABC\).
2. Middle right: Matches the 90° counterclockwise rotation orientation (sides and vertex order align with rotation).
3. Bottom right: Upright, same as original - not rotated.

So the correct \(\triangle DEF\) is the middle - right triangle (the one with \(F\) on the left, \(E\) on the top, \(D\) on the bottom - right among the three right - hand triangles).