Question

Construct a quadrilateral ABCD in which AB=4 cm, BC=3 cm, CD=4 cm, DA=3 cm and diagonal AC=5cm. Measure diagonal BD. Is it equal to AC? Find the measure of $\angle A$?

Answer 1

Hint: First, before proceeding for this, we must draw the required quadrilateral to get the idea of the actual figure it is representing. Then, we can clearly see from the figure that it represents a rectangle as its opposite sides are equal and one side is larger than the other. Then, by using the property of the rectangles, we know that all the angles of the rectangle is always ${{90}^{\circ }}$. Then, by using the Pythagoras theorem, we get the length of the diagonal BD.

Complete step by step answer:
In this question, we are supposed to find the length of diagonal BD and measure of $\angle A$ when a quadrilateral ABCD in which AB=4 cm, BC=3 cm, CD=4 cm, DA=3 cm and diagonal AC=5cm is given.
So, before proceeding for this, we must draw the required quadrilateral to get the idea of the actual figure it is representing as:

So, we can clearly see from the figure that it represents a rectangle as its opposite sides are equal and one side is larger than the other.
So, by using the property of the rectangles, we know that all the angles of the rectangle is always ${{90}^{\circ }}$.
So, we get the value of the $\angle A$as ${{90}^{\circ }}$.
Then, by using the other property of the right angles triangle ABD as $\angle A$is ${{90}^{\circ }}$and AD acts like the perpendicular and AB acts like base.
Now, by using the Pythagoras theorem for the right angles triangle ABD from the figure which states that square of the hypotenuse h is equal to the sum of the square of the base b and perpendicular p as:
${{h}^{2}}={{b}^{2}}+{{p}^{2}}$
Now, by substituting the values of the h as BD, p as 3 cm and b as 4 cm from the triangle ABD, we get:
$\begin{align}
  & B{{D}^{2}}={{4}^{2}}+{{3}^{2}} \\
 & \Rightarrow B{{D}^{2}}=16+9 \\
 & \Rightarrow B{{D}^{2}}=25 \\
 & \Rightarrow BD=\sqrt{25} \\
 & \Rightarrow BD=5cm \\
\end{align}$
So, we get the length of the diagonal BD as 5 cm.
Moreover, we can also conclude that the length of the diagonals of the rectangle are always equal and it is also proved by the condition given in the question as AC=5cm.
Hence, the length of the diagonal BD is 5cm and it is equal to the diagonal AC and the measure of $\angle A$is ${{90}^{\circ }}$.
Note:
Now, to solve these types of questions we need to know some of the basics of the figure which rectangle in this question. So, we need not solve for the length of the diagonal BD as there is a property of rectangles that its both diagonal lengths are always equal.