7.6.7 Check Your Understanding \( \triangle N T L \) is a triangle where \( m \angle N T L=90^{\circ}, m \angle T L N=45^{\circ} \) and \( N T=\frac{\sqrt{3}}{2} \). What is the length of \( \overline{T L} \) ? o \( \sqrt{3} \) o \( \frac{\sqrt{3}}{2} \) o \( \frac{\sqrt{6}}{2} \) o \( \sqrt{6} \)

To determine the length of \( \overline{TL} \) in triangle \( \triangle NTL \), let's analyze the given information: 1. **Triangle Properties:** - \( m \angle NTL = 90^\circ \): This means vertex \( T \) is the right angle. - \( m \angle TLN = 45^\circ \): This implies that vertex \( L \) has an angle of \( 45^\circ \). - Since the sum of angles in a triangle is \( 180^\circ \), the remaining angle at vertex \( N \) must also be \( 45^\circ \). 2. **Type of Triangle:** - With angles \( 90^\circ \), \( 45^\circ \), and \( 45^\circ \), \( \triangle NTL \) is a **45-45-90** right triangle. This type of triangle is **isosceles**, meaning the two legs are equal in length. 3. **Given Side Length:** - \( NT = \frac{\sqrt{3}}{2} \) 4. **Finding \( TL \):** - In a 45-45-90 triangle, the legs opposite the \( 45^\circ \) angles are equal. Therefore: \[ TL = NT = \frac{\sqrt{3}}{2} \] 5. **Verification with Options:** - The correct answer matches one of the provided options. **Answer:** \( \frac{\sqrt{3}}{2} \)