Question

△lmn is rotated about point p to create the image △lmn. ( mangle lpl = 105^circ ), ( mangle lmn = 34^circ ), and ( ln = 2.5 ) centimeters. use the definition of a rotation to find the following measures. ( mangle lmn = ) ( square^circ ), ( ln = square ) cm, ( mangle npn = square^circ )

Explanation:

Step1: Recall rotation properties (angle)

A rotation is a rigid transformation, so corresponding angles of the pre - image and image are equal. In $\triangle LMN$ and $\triangle L'M'N'$, $\angle LMN$ and $\angle L'M'N'$ are corresponding angles. Given $m\angle LMN = 34^{\circ}$, so $m\angle L'M'N'=m\angle LMN = 34^{\circ}$.

Step2: Recall rotation properties (side)

In a rotation (rigid transformation), corresponding sides of the pre - image and image are equal. The side $LN$ in $\triangle LMN$ and the side $L'N'$ in $\triangle L'M'N'$ are corresponding sides. Given $L'N' = 2.5$ cm, so $LN=L'N' = 2.5$ cm.

Step3: Recall rotation properties (angle of rotation)

The angle of rotation for a point about the center of rotation $P$ is the same for all corresponding points. The angle $\angle LPL'$ is the angle of rotation, and $\angle NPN'$ is also the angle of rotation (since $L$ and $N$ are both rotated about $P$). Given $m\angle LPL'=105^{\circ}$, so $m\angle NPN' = 105^{\circ}$.

Answer:

$m\angle L'M'N'=\boldsymbol{34}$°
$LN=\boldsymbol{2.5}$ cm
$m\angle NPN'=\boldsymbol{105}$°