Question

In the adjoining figure, altitudes YZ and XT of ∆WXY intersect at P. Prove that, 

i. □ WZPT is cyclic. 

ii. Points X, Z, T, Y are concyclic. 

Answer 1

Given: seg YZ ⊥ side

seg XT ⊥ side WY To

 To prove: i. □WZPT is cyclic.

ii. Points X, Z, T, Y are concyclic.

Proof:

 i. segYZ ⊥ side XW [Given] 

∴∠PZW = 90°…… (i) seg XT I side WY [Given] 

∴ ∠PTW = 90° ……(ii) 

∠PZW + ∠PTW = 90° + 90° [Adding (i) and (ii)] 

∴∠PZW + ∠PTW = 180° ∴□WZPT Ls a cyclic quadrilateral. [Converse of cyclic quadrilateral theorem] 

ii. ∠XZY = ∠YTX = 90° [Given]

∴ Points X and Y on line XY subtend equal angles on the same side of line XY. 

∴ Points X, Z, T and Y are concydic. [If two points on a given line subtend equal angles at two distinct points which lie on the same side of the line, then the four points are concyclic]