Question

the length of segment xy is 9 cm. which statements regarding triangle xyz are correct? select two options.
□ yz = 9 cm
□ xz = 9 cm
□ xz = 9√2 cm
□ xz = 2(xy)
□ yz is the longest segment in △xyz.
(image: right triangle xyz with right angle at y, xy = 9 cm, ∠x = 45°, ∠z = 45°)

Explanation:

Step1: Identify triangle type

Triangle \( XYZ \) is a right - isosceles triangle (since \( \angle Y = 90^{\circ}\), \( \angle X=45^{\circ}\), \( \angle Z = 45^{\circ}\)), so \( XY = YZ \). Given \( XY = 9\space cm \), then \( YZ=9\space cm \).

Step2: Calculate hypotenuse \( XZ \)

In a right - isosceles triangle with legs of length \( a \), the hypotenuse \( c=a\sqrt{2}\). Here \( a = XY=9\space cm \), so \( XZ = 9\sqrt{2}\space cm \). Also, check other options: \( XZ
eq9\space cm \), \( XZ
eq2(XY) \) (since \( 2(XY)=18\space cm \) and \( 9\sqrt{2}\approx12.73\space cm\)), and the longest segment is \( XZ \) (hypotenuse), not \( YZ \).

Answer:

YZ = 9 cm, XZ = \( 9\sqrt{2}\space cm \) (i.e., the options "YZ = 9 cm" and "XZ = \( 9\sqrt{2}\space cm \)")