Question

In the adjoining equilateral triangle PQR , X , Y and Z are the middle points of the sides PQ , QR and RP respectively. Prove that XYZ is also an equilateral triangle.

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Answer 1

this is your answer. thanks, for great point

Answer 2

See Below

Explanation:

Statements: Reasons:

`1)\text{ } \Delta PQR\text{ is equilateral}` Given

`2)\text{ }PQ=QR=RP` Definition of Equilateral

`3)\text{ } X, Y, Z\text{ are the midpoints of } PQ, QR, RP` Given

`4)\text{ } PX=XQ` Definition of Midpoint

`5)PQ=PX+XQ` Segment Addition

`6)\text{ } PQ=2XQ` Substitution

`7)\text{ } QY=YR` Definition of Midpoint

`8)\text{ }QR=QY+YR` Segment Addition

`9)\text{ } QR=2QY` Substitution

`10)\text{ } 2XQ=2QY` Substitution

`11)\text{ } XQ=QY` Division Property of Equality

`12)\text{ } XQ=PX=QY=YR` Transitive Property

`13)\text{ } RZ=ZP` Definition of Midpoint

`14)\text{ } RP=RZ+ZP` Segment Addition

`15)\text{ } RP=2RZ` Substitution

`16)\text{ } 2XQ=2RZ` Substitution

`17)\text{ } XQ=RZ` Substitution

`18)\text{ } XQ=PX=QY=YR=RZ=ZP` Transitive Property

`19)\text{ } \angle Q\cong \angle R\cong \angle P` Definition of Equilateral

`20)\text{ } \Delta XQY\cong \Delta YRQ \cong \Delta ZPX` SAS Congruence

`21)\text{ } XY\cong YZ\cong ZX` CPCTC

`22)\text{ } \Delta XYZ\text{ is equilateral}` Equilateral Triangle Theorem