For isosceles triangle, two equal side is of 20 cm. and angle between them is 30 degree. Find the area of the traingle.
Without applying trigonometry
Answers
Topic :-
Triangles
Given :-
For an isosceles triangle, two equal side is of 20 cm and angle between them is 30°.
To Find:-
Area of the given triangle without applying trigonometry.
Solution :-
Let us consider an isosceles triangle ABC in which AB = BC = 20 cm and ∠ABC = 30°.
- Draw AD ⊥ BC.
- Extend AD to point E such that AD = DE.
- Jôin B and E.
In ΔABD and ΔEBD, we can observe that,
AD = ED (by construction)
∠ADB = ∠EDB (both are right angles)
BD = BD (common side)
Hence from SAS rule of congruency, we can say that ΔABD and ΔEBD are congruent.
ΔABD ≅ ΔEBD
∴ ∠ABD = ∠EBD = 30°
Now,
∠ABE = ∠ABD + ∠EBD
∠ABE = 30° + 30°
∠ABE = 60°
∠DAB + ∠ABD + ∠BDA = 180°
∠DAB + 30° + 90° = 180°
∠DAB + 120° = 180°
∠DAB = 180° - 120°
∠DAB = 60°
∠DEB + ∠EBD + ∠BDE = 180°
∠DEB + 30° + 90° = 180°
∠DEB + 120° = 180°
∠DEB = 180° - 120°
∠DEB = 60°
∴ ∠ABE = ∠DAB = ∠DEB = 60°
All interior angles of ΔABE = 60°.
Hence, ΔABE is an equilateral triangle.
∴ AB = BE = AE
∴ 20 cm = BE = AE
AE = AD + DE
AE = AD + AD
(∵ AD = DE by construction)
20 cm = 2(AD)
AD = 20 cm / 2 = 10 cm
Area of a triangle = (0.5) × Base × Height
Area of triangle ABC = (0.5) × BC × AD
Area of triangle ABC = (0.5) × 20 cm × 10 cm
Area of triangle ABC = (0.5) × 200 sq. cm
Area of triangle ABC = 100 sq. cm
Answer :-
So, area of given isosceles triangle is 100 sq. cm.
Answer:
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