Maths-
General
Easy

Question

A framed television has a ratio of width to height of about 1.732 : 1. For a television of height h inches, what is an equivalent expression for the length of the diagonal? Justify your answer. Write an expression for parimeter.

The correct answer is: Length of the diagonal for the given framed tv is 2h units where h is its height and its perimeter can be expressed in terms of its height as- 5.5h units2


    Hint:-
    1. By Pythagorean theorem, for a right angled triangle-
    (Hypotenuse)2 = Sum of the squares of other 2 sides …..................... (Where Hypotenuse is the side opposite to the right angle)
    2. Perimeter of a rectangle = 2 (length + breadth)

    Step-by-step solution:-
    Let the length, width, height and diagonal for the given framed tv be l, w, h & d, respectively.
    From the given information, we get-
    Width / Height = 1.732 / 1
    ∴ 1 (Width) = 1.732 (Height)
    ∴ Width = 1.732 × Height
    ∴ w = 1.732 h ….................................................... (Equation i)
    Now, from the adjacent diagram, we can observe that-
    Diagonal cuts the framed tv into 2 triangles with a right angle (as edges of a rectangle form right angle)
    ∴ using Pythagorean theorem, we get-
    (Diagonal)2 = (Width)2 + (Height)2
    ∴ d2 = (1.732 h)2 + h2 .................................................... (From Equation i)
    ∴ d2 = 3h2 + h2
    ∴ d2 = 4h2
    ∴ d = 2h ........................................................................ (Taking square roots both the sides)
    Perimeter of the given framed tv = 2 (width + height)
    ∴ Perimeter of the given framed tv = 2 (1.732h + h) ................. (From Equation i)
    ∴ Perimeter of the given framed tv = 2 × 2.732h
    ∴ Perimeter of the given framed tv ≈ 5.5 h units2.
    Final Answer:-
    ∴ Length of the diagonal for the given framed tv is 2h units where h is its height and its perimeter can be expressed in terms of its height as- 5.5h units2.

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