Question

the graph shows quadrilaterals fghi and lmno.
is fghi congruent to lmno? justify your answer.
yes, because a rotation 90° clockwise around the origin maps defg onto lmno.
yes, because a translation right 1 unit and up 11 units maps fghi onto lmno.

Explanation:

Step1: Check the side - lengths

Use the distance formula \(d=\sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}\).
For example, for \(FGHI\):
Let \(F(- 9,-7)\), \(G(-5,-3)\). Then \(d_{FG}=\sqrt{(-5 + 9)^2+(-3 + 7)^2}=\sqrt{16 + 16}=\sqrt{32}=4\sqrt{2}\).
For \(LMNO\):
Let \(L(-7,9)\), \(M(-3,5)\). Then \(d_{LM}=\sqrt{(-3 + 7)^2+(5 - 9)^2}=\sqrt{16 + 16}=\sqrt{32}=4\sqrt{2}\).

Step2: Check the transformation

A translation is a rigid transformation. A translation \((x,y)\to(x + h,y + k)\).
If we consider the translation of \(F(-9,-7)\) to \(O(-8,4)\): \(x\) - coordinate: \(-9+1=-8\), \(y\) - coordinate: \(-7 + 11 = 4\).
For \(G(-5,-3)\): \(x=-5 + 1=-4\), \(y=-3+11 = 8\) (not correct).
A rotation is a rigid transformation.
The rotation rule for a \(90^{\circ}\) clockwise rotation about the origin is \((x,y)\to(y,-x)\).
For \(F(-9,-7)\): after rotation \((-7,9)\) (matches \(L\)).
For \(G(-5,-3)\): after rotation \((-3,5)\) (matches \(M\)).
For \(H(-3,-5)\): after rotation \((-5,3)\) (matches \(N\)).
For \(I(-6,-8)\): after rotation \((-8,6)\) (matches \(O\)).

Answer:

Yes, because a rotation \(90^{\circ}\) clockwise around the origin maps \(FGHI\) onto \(LMNO\).