In the Given Figure, Altitudes Yz and Xt of ∆Wxy Intersect at P. Prove That,(1) ▢Wzpt is Cyclic.(2) Points X, Z, T, Y Are Concyclic. - Geometry Mathematics 2

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Sum

In the given figure, altitudes YZ and XT of ∆WXY intersect at P. Prove that,
(1) ▢WZPT is cyclic.
(2) Points X, Z, T, Y are concyclic.

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Solution

(1) It is given that, YZ ⊥ WX and XT ⊥ WY.
∴ ∠WZY = 90º     .....(1)
∠WTX = 90º         .....(2)
Adding (1) and (2), we get
∠WZY + ∠WTX = 90º + 90º = 180º
Or ∠WZP + ∠WTP = 90º + 90º = 180º
In quadrilateral WZPT,
∠WZP + ∠WTP = 180º​
We know, if a pair of opposite angles of a quadrilateral is supplementary, then quadrilateral is cyclic.
Therefore, quadrilateral WZPT is cyclic.
(2) It is given that, YZ ⊥ WX and XT ⊥ WY.
∴ ∠XZY = 90º and ∠XTY = 90º
 ⇒ ∠XZY = ∠XTY
So, two points X and Y on the line XY subtends equal angles at two distinct points Z and T which lie on the same side of the line XY.
Therefore, the points X, Z, T and Y are concyclic.

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Chapter 3: Circle - Practice Set 3.4 [Page 74]
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