The base of an isosceles triangle is equal to $$4$$ units, the base angle is equal to $$450$$. A straight line cuts the extension of the base at a point M at the angle θ and bisects the lateral side of the triangle which is nearest to M.The area of quadrilateral which the straight line cuts off from the given triangle isThe possible range of values in which area of quadrilateral which straight line cuts off from the given triangle lie inThe length of portion of straight line inside the triangle may lie in the range

Question

The base of an isosceles triangle is equal to $$4$$ units, the base angle is equal to $$450$$. A straight line cuts the extension of the base at a point M at the angle θ and bisects the lateral side of the triangle which is nearest to M.The area of quadrilateral which the straight line cuts off from the given triangle isThe possible range of values in which area of quadrilateral which straight line cuts off from the given triangle lie inThe length of portion of straight line inside the triangle may lie in the range

Infinity Learn · Accepted Answer

Equation of line PMy−$$1=$$$$tan$$θ(x$$−$$1)Intersection$$ point Q of AC and $$MP4$$−x−$$1=$$$$tan$$θ$$(x$$−$$1)⇒Q≡$$3+$$$$tan$$θ$$1+$$$$tan$$θ,$$1+3tan$$θ$$1+$$$$tan$$θArea of Δ$$APQ=modulus$$ $$of121112213+$$$$tan$$θ$$1+$$$$tan$$θ$$1+3tan$$θ$$1+$$$$tan$$θ$$1=1$$−$$tan$$θ$$1+$$$$tan$$θArea of quadrilateral $$BPQCA=12$$×$$4$$×$$2$$−$$1$$−$$tan$$θ$$1+$$$$tan$$θ$$=3+5tan$$θ$$1+$$$$tan$$θ$$A=5$$−$$21+$$$$tan$$θθ$$=(0$$,π$$4)$$⇒$$1+$$$$tan$$θ∈(1$$,$$2)⇒A∈(3$$,$$4)(PQ)2=3+$$$$tan$$θ$$1+$$$$tan$$θ−$$12+1+3tan$$θ$$1+$$$$tan$$θ−$$124(1+tan2$$θ$$)1+tan2$$θ$$+2tan$$θ$$=41+sin2$$θ⇒$$(PQ)2$$∈$$(2$$,$$4)⇒PQ∈(2$$,$$2)

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