Answer:
1. PR = 8 units
2. ST = 5 units
Step-by-step explanation:
1. Consider triangles QSP and RSP. In these triangles,
[tex]\angle QSP\cong \angle RSP[/tex] - all right angles are congruent
[tex]PS\cong PS[/tex] - reflexive property
[tex]QS\cong RS[/tex] - given
So, [tex]\triangle QSP\cong \triangle RSP[/tex] by SAS postulate (SAS postulate states that if two sides and the included angle of one triangle are congruent to two sides and the included angle of another triangle, then these two triangles are congruent).
Congruent triangles have congruent corresponding parts, then
[tex]PQ\cong PR[/tex]
Given [tex]PQ=8,\ \ PR=2x,[/tex] you get
[tex]2x=8\\ \\x=4[/tex]
and
[tex]PR=2\cdot 4=8\ units[/tex]
2. Consider triangles VST and VUT. In these triangles,
[tex]\angle VST\cong \angle VUT[/tex] - all right angles are congruent
[tex]VT\cong VT[/tex] - reflexive property
[tex]\angle SVT\cong \angle UVT[/tex] - given
So, [tex]\triangle SVT\cong \triangle UVT[/tex] by AAS postulate (AAS postulate states that if two angles and the non-included side one triangle are congruent to two angles and the non-included side of another triangle, then these two triangles are congruent).
Congruent triangles have congruent corresponding parts, then
[tex]ST\cong UT[/tex]
Given
[tex]ST=5z\\ \\TU=2z+3,[/tex]
you get
[tex]5z=2z+3\\ \\5z-2z=3\\ \\3z=3\\ \\z=1[/tex]
Then
[tex]ST=5\cdot 1=5\ units[/tex]
