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Q1(b):
Find the area of each of the following parallelograms:
(b) 
Find the area of each of the following parallelograms:
(b)
Solution :
Step 1: Given Variables & Initial Setup
Based on the standard geometric parameters provided in the referenced figure for this problem, we extract the primary dimensions of the parallelogram. Let the parallelogram be denoted as $ABCD$.
- Base ($b$): The length of the bottom edge of the parallelogram is given as $5\text{ cm}$.
- Height ($h$): The perpendicular distance (altitude) from the opposite parallel side to the base is given as $3\text{ cm}$.
Step 2: Theoretical Foundation & Formula
The area of a parallelogram is defined as the total two-dimensional space enclosed within its four sides. It is calculated by taking the product of its base and its corresponding perpendicular height.
Formula:
$\text{Area of a Parallelogram} = \text{Base} \times \text{Height}$
[Justification: By the principle of geometric decomposition, a right-angled triangle can be conceptually "cut" from one end of the parallelogram and translated to the opposite end. This transformation converts the parallelogram into a rectangle of identical base and height, proving that they share the same area formula.]
Step 3: Geometric Visualization
Below is a mathematically scaled representation of the parallelogram $ABCD$, where the base $AB = 5\text{ cm}$ and the altitude $DE = 3\text{ cm}$. The drawing maintains a strict $40:1$ coordinate ratio to ensure absolute visual accuracy.
Step 4: Execution of Area Calculation
We substitute the given scalar values into the area formula. Ensure that both measurements are in the same units (centimeters) before multiplying to yield an area in square centimeters ($\text{cm}^2$).
$\text{Area} = b \times h$
$\text{Area} = 5\text{ cm} \times 3\text{ cm}$
$\text{Area} = (5 \times 3) \text{ cm}^{1+1}$
$\text{Area} = 15\text{ cm}^2$
Final Solution: The area of the given parallelogram is $15\text{ cm}^2$.
More Questions from Class 9 Mathematics Coordinate Geometry EXERCISE 9.1
- Q1(a): Find the area of each of the following parallelograms: (a)
- Q1(c): Find the area of each of the following parallelograms: (c)
- Q1(d): Find the area of each of the following parallelograms: (d)
- Q1(e): Find the area of each of the following parallelograms: (e)
- Q2(a): Find the area of each of the following triangles: (a)
- Q2(b): Find the area of each of the following triangles: (b)
- Q2(c): Find the area of each of the following triangles: (c)
- Q2(d): Find the area of each of the following triangles: (d)
- Q3(a): Find the missing values: Base = $20$ cm, Height = ______, Area of the Parallelogram = $246$ cm$^2$.
- Q3(b): Find the missing values: Base = ______, Height = $15$ cm, Area of the Parallelogram = $154.5$ cm$^2$.
- Q3(c): Find the missing values: Base = ______, Height = $8.4$ cm, Area of the Parallelogram = $48.72$ cm$^2$.
- Q3(d): Find the missing values: Base = $15.6$ cm, Height = ______, Area of the Parallelogram = $16.38$ cm$^2$.
- Q4(a): Find the missing values: Base = $15$ cm, Height = ______, Area of Triangle = $87$ cm$^2$.
- Q4(b): Find the missing values: Base = ______, Height = $31.4$ mm, Area of Triangle = $1256$ mm$^2$.
- Q4(c): Find the missing values: Base = $22$ cm, Height = ______, Area of Triangle = $170.5$ cm$^2$.
- Q5(a): $PQRS$ 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: (a) the area of the parallegram $PQRS$.
- 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.
- Q6: $DL$ and $BM$ are the heights on sides $AB$ and $AD$ respectively of parallelogram $ABCD$ (Fig 9.15). If the area of the parallelogram is $1470$ cm$^2$, $AB = 35$ cm and $AD = 49$ cm, find the length of $BM$ and $DL$.
- Q7: $\triangle ABC$ is right angled at $A$ (Fig 9.16). $AD$ is perpendicular to $BC$. If $AB = 5$ cm, $BC = 13$ cm and $AC = 12$ cm, Find the area of $\triangle ABC$. Also find the length of $AD$.
- Q8: $\triangle ABC$ is isosceles with $AB = AC = 7.5$ cm and $BC = 9$ cm (Fig 9.17). The height $AD$ from $A$ to $BC$, is $6$ cm. Find the area of $\triangle ABC$. What will be the height from $C$ to $AB$ i.e., $CE$?
CBSE Solutions for Class 9 Mathematics Coordinate Geometry
Chapters in CBSE - Class 9 Mathematics
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