ABCD is a square with side 12 cm and E is a point in the interior of ABCD if angle EDC =angle ECD =15 degree then find the perimeter of triangle AEB
Answers
Given : ABCD is a square with side 12 cm and E is a point in the interior of ABCD if angle EDC =angle ECD =15 degree
To find : perimeter of triangle AEB
Step-by-step explanation:
Draw a line segment PQ ║ AD & BC passing through E and intersecting CD at P & AB at Q
PQ = 12 cm
PQ ⊥ DC & AB
P will be mid point of CD as angle EDC =angle ECD
=> ECD is an isosceles triangle & median & altitude coincide
=> DP = CP = CD/2 = 12/2 = 6 cm
Tan ∠ECP = PE/CP
Tan 15° = PE/6
=> PE = 6 Tan 15°
QE = PQ - PE = 12 - 6 Tan 15°
AQ = BQ = AB/2 = 6 cm
BE² = AE² = 6² + ( 12 - 6 Tan 15°)²
= 36 + 144 + 36Tan²15° - 144 Tan 15°
= 144 + 36 + 36Sin²15°/Cos²15° - 144Sin15°/Cos15°
= 144 + ( 36 Cos²15° + 36 Sin²15° - 144Sin15°Cos15°)
= 144 + ( 36(Cos²15° + Sin²15° ) - 72* 2Sin15°Cos15°)
= 144 + ( 36(1 ) - 72* Sin30°)
= 144 + ( 36 - 72* (1/2))
= 144 + ( 36 - 36)
= 144
BE² = AE² = 144
=> BE = AE = 12
perimeter of triangle AEB = AE + BE + AB
= 12 + 12 + 12
= 36 cm
36 cm is the perimeter of triangle AEB
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perimeter of triangle AEB
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