The bisector of the base angles Q and R of an equilateral triangle PQR meet at S. ST and SM are drawn parallel to the sides PQ and PR. Show that QT=TM=MR

From TM = MS, MS = MR, and TM = MR

QT = TM = MR Transitive property

Step-by-step explanation:

ΔPQR is an equilateral triangle Given

∠PQR = ∠RPQ = ∠PRQ = 60° Definition of equilateral triangle

∠PQR = ∠SQT + ∠SQP = 60° Addition property of angles

∠PRQ = ∠SRM + ∠SRP = 60° Addition property of angles

In triangle PQR, ∠SQT = ∠SQP = 30° = ∠SRM = ∠SRP Angles bisected by angle bisector

∠STM = ∠PQR - 60° Angle of same side of two parallel lines crossing the same transversal

∠PRQ = ∠SMT = 60° Angle of same side of two parallel lines crossing the same transversal

∠MST = 180 - ∠SMT - ∠STM = 180 - 60 - 60 = 60°

∠MST = ∠SMT = ∠STM = 60°, ΔMST is an equilateral triangle, all angles equal

ST = MS = TM definition of equilateral triangle

∠STQ = 180 - ∠STM = 180° - 60° = 120° (∠STQ and ∠STM are supplementary angles)

∠QST = 180 - ∠SQT - ∠STQ = 180 - 30 - 120 = 30 (Interior angles in the same triangle)

ΔQTS is an  isosceles triangle, base angles are equal

∴ ST = QT Definition of equilateral triangle

Similarly

∠SMR = 180 - ∠SMT = 180° - 60° = 120° ∠SMR and ∠SMT are supplementary angles

∠MSR = 180 - ∠SRM - ∠SMR = 180 - 30 - 120 = 30 (Angles in the same triangle)

ΔRMS is an  isosceles triangle, base angles are equal

∴ MS = MR Definition of equilateral triangle

Since ST = QT, and,  ST = TM, QT = TM Transitive property

Also since TM = MS, and MS = MR, TM = MR Transitive property

Therefore;

QT = TM = MR Transitive property

bn,mn,n.vn.

step-by-step explanation:

maximum area = xy = (16)(32) = 512 ft2

to maximize the area of the garden, the side of the fence parallel to the side of the house should be 32 ft long, and the other two sides should both have length 16 ft.'

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