Share

## Given: Circle k(O) with OT ⊥ XY, and OU ⊥ WZ and X=Z. Prove: XY=WZ

Question

Given: Circle k(O) with OT ⊥ XY, and OU ⊥ WZ and X=Z. Prove: XY=WZ

in progress
0

Mathematics
7 months
2021-08-09T19:51:03+00:00
2021-08-09T19:51:03+00:00 1 Answers
2 views
0
## Answers ( )

ΔXOY ≅ ΔZOW ⇒ proved down

Step-by-step explanation:

* Lets study some facts on the circle

– If two chords equidistant from the center of the circles,

then they are equal in length

* the meaning of equidistant is the perpendicular distances

from the center of the circle to the chords are equal in length

* Lets check this fact in our problem

∵ XY and WZ are two chords in circle O

∵ OT ⊥ XY

– OT is the perpendicular distance from the center to the chord XY

∵ OU ⊥ WZ

– OU is the perpendicular distance from the center to the chord WZ

∵ OT ≅ OU

– The two chords equidistant from the center of the circle

∴ The two chords are equal in length

∴ XY ≅ WZ

* Now in the two triangles XOY and ZOW , to prove that

they are congruent we must find one of these cases:

1- SSS ⇒ the 3 sides of the 1st triangle equal the corresponding

sides in the 2nd triangle

2- SAS ⇒ the two sides and the including angle between them

in the 1st triangle equal to the corresponding sides and

including angle in the 2nd triangle

3- AAS ⇒ the two angles and one side in the 1st triangle equal the

corresponding angles and side in the 2nd triangle

* Lets check we will use which case

– In the two triangles XOY and ZOW

∵ XY = ZW ⇒ proved

∵ OX = OZ ⇒ radii

∵ OY = OW ⇒ radii

* This is the first case SSS

∴ ΔXOY ≅ ΔZOW