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Q5(b):

$PQRS$ is a parallelogram (Fig 9.14). $QM$ is the height from $Q$ to $SR$ and $QN$ is the height from $Q$ to $PS$. If $SR = 12$ cm and $QM = 7.6$ cm. Find: (b) $QN$, if $PS = 8$ cm.

Solution :

Given Variables & Initial Setup

We are analyzing the parallelogram $PQRS$ with two distinct base-height pairings. The geometric properties provided are:

  • Base $SR = 12 \text{ cm}$
  • Height corresponding to base $SR$, denoted as $QM = 7.6 \text{ cm}$
  • Base $PS = 8 \text{ cm}$
  • Height corresponding to base $PS$, denoted as $QN$, which is the unknown variable to be determined.
P Q R S M N 12 cm 7.6 cm 8 cm QN = ?

Step 1: Calculating the Area of Parallelogram $PQRS$

The area of a parallelogram is defined as the product of any chosen base and its corresponding perpendicular height. [Per the geometric theorem for the area of a parallelogram: $\text{Area} = \text{base} \times \text{height}$].

Using the given base $SR$ and its corresponding height $QM$:

$\text{Area of } PQRS = SR \times QM$
$\text{Area of } PQRS = 12 \text{ cm} \times 7.6 \text{ cm}$
$\text{Area of } PQRS = 91.2 \text{ cm}^2$

Step 2: Formulating the Equation for Height $QN$

The area of a specific polygon is an invariant scalar quantity; it remains constant regardless of which side is designated as the base. [By the principle of area invariance]. Therefore, calculating the area using base $PS$ and its corresponding height $QN$ must yield the exact same result.

$\text{Area of } PQRS = PS \times QN$

Substituting the known area ($91.2 \text{ cm}^2$) and the length of base $PS$ ($8 \text{ cm}$) into the equation:

$91.2 = 8 \times QN$

Step 3: Solving for $QN$

To isolate $QN$, we divide both sides of the equation by $8$:

$QN = \frac{91.2}{8}$

Performing the division:

$QN = 11.4 \text{ cm}$

Final Solution: The length of the height $QN$ is $11.4 \text{ cm}$.


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