Question

1. what is the length of segment su?

16 cm
8 cm
16√2cm
16cm
8cm
8√2cm

Explanation:

Step1: Identify triangle type

Triangle \( SVU \) is a right - isosceles triangle (since one angle is \( 45^{\circ} \) and \( \angle VUS = 90^{\circ} \), so the other non - right angle is also \( 45^{\circ} \)). In a right - isosceles triangle, the legs are equal.

Step2: Determine leg length

We know that one leg \( VU = 8\space cm \). In a right - isosceles triangle with legs of length \( a \), the hypotenuse \( SU \) can be found using the Pythagorean theorem \( SU=\sqrt{VU^{2}+SU_{leg}^{2}} \), but since it's isosceles (\( VU = SU_{leg} = 8\space cm \)), or we can use the property of \( 45 - 45-90 \) triangles where the hypotenuse \( c=a\sqrt{2} \), where \( a \) is the length of a leg. Here \( a = 8\space cm \), so \( SU = 8\sqrt{2}\space cm \)? Wait, no, wait. Wait, in triangle \( SVU \), \( \angle SVU=45^{\circ} \), \( \angle VUS = 90^{\circ} \), so \( \angle USV = 45^{\circ} \), so \( VU=SU \)? Wait, no, \( VU \) is one leg, \( SU \) is the other leg? Wait, no, looking at the diagram, \( VU = 8\space cm \), and triangle \( SVU \) has angles \( 45^{\circ},90^{\circ},45^{\circ} \), so the two legs are equal. Wait, but maybe I made a mistake. Wait, let's re - examine. Wait, the triangle \( RVU \): \( RV = 16\space cm \), \( \angle VR S=60^{\circ} \), but maybe triangle \( SVU \) is isosceles right - angled with \( VU = 8\space cm \), so \( SU = VU = 8\space cm \)? No, that can't be. Wait, no, in a \( 45 - 45-90 \) triangle, if the leg is \( a \), hypotenuse is \( a\sqrt{2} \), but if the two legs are equal. Wait, maybe the triangle \( SVU \) has \( VU = 8\space cm \) and \( \angle SVU = 45^{\circ} \), \( \angle VUS=90^{\circ} \), so \( SU = VU = 8\space cm \)? No, that's not right. Wait, no, let's calculate the angle at \( V \) in triangle \( RVV \) (wait, no, triangle \( RVU \)). Wait, \( RV = 16\space cm \), maybe we can find \( \angle RVU \). In triangle \( RVU \), \( \angle R = 60^{\circ} \), \( \angle RVU = 90^{\circ} \), so \( \angle RUV=30^{\circ} \). Then, the length of \( RU \) can be found, but maybe that's not necessary. Wait, the triangle \( SVU \): \( \angle SVU = 45^{\circ} \), \( \angle VUS = 90^{\circ} \), so \( \angle USV = 45^{\circ} \), so it's an isosceles right triangle with legs \( VU \) and \( SU \). So if \( VU = 8\space cm \), then \( SU = 8\space cm \)? No, that's conflicting. Wait, no, maybe I misread the triangle. Wait, the segment \( VU \) is \( 8\space cm \), and \( SU \) is perpendicular to \( VT \), \( \angle SVU = 45^{\circ} \), so in right triangle \( SVU \), \( \tan(45^{\circ})=\frac{SU}{VU} \), since \( \tan(45^{\circ}) = 1 \), then \( SU = VU=8\space cm \)? But that's one of the options. Wait, the options are \( 16\sqrt{2}\space cm \), \( 16\space cm \), \( 8\space cm \), \( 8\sqrt{2}\space cm \). Wait, if \( \angle SVU = 45^{\circ} \), \( \angle VUS = 90^{\circ} \), then triangle \( SVU \) is isosceles right - angled, so \( SU = VU = 8\space cm \).

Answer:

\( 8\space cm \) (corresponding to the option with \( 8\space cm \))