Question

A rectangle is $14{\text{cm}}$ wide and $48{\text{cm}}$ long. How do you find the measure of the acute angle between its diagonals?

Answer 1

Hint:
In this question, we need to find the acute angle value which is formed between the diagonals of the rectangle. Firstly, we must know the basic properties related to rectangles, so that it helps to understand the problem clearly. Then find the measure of the acute angle formed between the diagonals using the formula, $2{\tan ^{ - 1}}\left( {\dfrac{{{\text{width}}}}{{{\text{length}}}}} \right)$.

Complete step by step solution:
Given a rectangle of width $14{\text{cm}}$ and length $48{\text{cm}}$.
We are asked to find out the measure of the acute angle formed between the diagonals.
Firstly, let us understand some basic properties related to the rectangle.
A rectangle is a polygon in which opposite sides are of equal length. And in the rectangle two sides which have longer length are called length of the rectangle and the two sides which have shorter length are called width of the rectangle.
The angle formed inside the rectangle at each side of the corner is equal to ${90^ \circ }$. i.e. the sides are perpendicular to each other.
Now we find the solution for the given problem.
Note that in a rectangle, when we draw a line between its diagonals we get two right angled triangles.
If we know the length and width of the rectangle, we have the opposite and adjacent sides of the right angled triangle that is formed.
So if we want to find the angle formed between the diagonals with respect to the adjacent side of the triangle we can use the tangent function, since the tangent function is related to the opposite and adjacent side.
We have the formula to find the acute angle formed between the diagonals of the rectangle, which is given by,
${\text{acute}}\,{\text{angle}} = 2{\tan ^{ - 1}}\left( {\dfrac{{{\text{width}}}}{{{\text{length}}}}} \right)$
In the given triangle width$ = $ $14{\text{cm}}$ and length $ = $ $48{\text{cm}}$
$ \Rightarrow 2{\tan ^{ - 1}}\left( {\dfrac{{14}}{{48}}} \right)$
$ \Rightarrow 2{\tan ^{ - 1}}(0.29167)$
$ \Rightarrow 2 \times {16.26^\circ } \approx {32.5^\circ }$

Hence the measure of the acute angle formed between the diagonals of rectangle of width $14{\text{cm}}$ and length $48{\text{cm}}$ is approximately given by ${32.5^\circ }$.

Note:
Students must know that in a rectangle opposite sides are of equal length and perpendicular to each other.
If we construct lines bisecting the two sides and the diagonals of a geometrical figure like rectangles, we can easily observe that the acute angle between the diagonals is twice the angle in a right angled triangle.
The formula for finding the acute angle formed between the diagonals of a rectangle is given by,
${\text{acute}}\,{\text{angle}} = 2{\tan ^{ - 1}}\left( {\dfrac{{{\text{width}}}}{{{\text{length}}}}} \right)$