Question

1. right triangle hij is shown graphed in the grid below with vertices at h(-8,3), i(-8,7), and j(-2,3).

(a) draw the image of △hij after a clockwise rotation of 90° about the origin. label the image △hij. show how the vertices are mapped below.
(b) draw the image of △hij after a reflection across the x - axis. label the image △hij. show how the vertices are mapped below.
(c) explain why △hij must be congruent to △hij.

Explanation:

(a)

Step1: Apply rotation formula

The formula for a clockwise rotation of \(90^{\circ}\) about the origin is \((x,y)\to(y, -x)\).
For \(H(-8,3)\):
\(x=-8,y = 3\), after rotation \(H'=(3,8)\)
For \(I(-8,7)\):
\(x=-8,y = 7\), after rotation \(I'=(7,8)\)
For \(J(-2,3)\):
\(x=-2,y = 3\), after rotation \(J'=(3,2)\)

(b)

Step1: Apply reflection formula

The formula for a reflection across the \(x -\)axis is \((x,y)\to(x,-y)\)
For \(H'(3,8)\):
\(x = 3,y=8\), after reflection \(H''=(3,-8)\)
For \(I'(7,8)\):
\(x = 7,y = 8\), after reflection \(I''=(7,-8)\)
For \(J'(3,2)\):
\(x = 3,y=2\), after reflection \(J''=(3,-2)\)

(c)

Step1: Use transformation properties

Rotations and reflections are rigid transformations. Rigid transformations preserve the shape and size of a figure. Since \(\triangle H'I'J'\) is obtained from \(\triangle HIJ\) by a rotation (a rigid transformation) and \(\triangle H''I''J''\) is obtained from \(\triangle H'I'J'\) by a reflection (a rigid transformation), \(\triangle HIJ\) and \(\triangle H''I''J''\) have the same side - lengths and angle measures.

Answer:

(a)

\(H(-8,3)\to H'(3,8)\), \(I(-8,7)\to I'(7,8)\), \(J(-2,3)\to J'(3,2)\)

(b)

\(H'(3,8)\to H''(3,-8)\), \(I'(7,8)\to I''(7,-8)\), \(J'(3,2)\to J''(3,-2)\)

(c)

Rotations and reflections are rigid transformations. Rigid - transformations preserve side - lengths and angle measures. So, \(\triangle HIJ\cong\triangle H''I''J''\)