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Find the measures of the angles between the diagonals of the rectangle whose vertices are a = (1, 0), b = (0, 3), c = (3, 4), and d = (4, 1)



Answer :

Answer:

The angle between the diagonal of the rectangle is [tex]\frac{\pi }{2}[/tex].

Given:

The vertices of the rectangle are

a = (1, 0), b = (0, 3), c = (3, 4), and d = (4, 1)

To find:

The objective is to find the angle between the diagonals.

Step 1 of 3

Consider the diagram attached.

Step 2 of3

The position vector of diagonal AC is-

=(3-1)î+(4-0)j=2î+4j

And the position vector of diagonal BD is-

=(4-0)î+(1-3)j=4î-2j

Step 3 of 3

cosθ=[tex]\frac{AC.BD}{|AC|.|BD|}[/tex]

cosθ=[tex]\frac{(2i-4j)(4i-2j)}{\sqrt{2^2+4^2} \sqrt{4^2+(-2)^2\\} }[/tex]

=[tex]\frac{8-8}{\sqrt{20}.\sqrt{20} }[/tex]

=0

θ=[tex]cos^{-1}[/tex](0)

θ=[tex]\frac{\pi }{2}[/tex]

Therefore the angle between the diagonal of the rectangle is [tex]\frac{\pi }{2}[/tex]

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varshamittal029

The angle between the diagonals of a rectangle is π/2

  • A rectangle is a parallelogram, so  opposite sides are equal. The diagonals of rectangle are equal and bisect each other at 90°.

  • A diagonal of a rectangle is a line segment that connects two non-adjacent vertices.

  • A diagonal divides the rectangle in 2 right triangles. where the sides are equal to the sides and the hypotenuse of the rectangle.

The vertices of the rectangle are

a = (1, 0), b = (0, 3), c = (3, 4),  d = (4, 1)

The diagonal position vector AC  

=(3-1)î+(4-0)j=2î+4j

And the diagonal position vector BD  

=(4-0)î+(1-3)j=4î-2j

cos Ф = (AC.BD) / (|AC| . |BD|)

cos Ф = (8-8) / (root20 -roo20)

= 0

Ф = π/2

Therefore the angle between the diagonals of the rectangle is π/2

Learn more about diagonals here

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