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Q3(c):
Find the missing values: Base = ______, Height = $8.4$ cm, Area of the Parallelogram = $48.72$ cm$^2$.
Solution :
Given Variables & Initial Setup
We are provided with the following geometric parameters for a parallelogram:
- Area ($A$) = $48.72 \text{ cm}^2$
- Height ($h$) = $8.4 \text{ cm}$
- Base ($b$) = ? (Unknown)
Step 1: Stating the Governing Geometric Formula
The area of a two-dimensional parallelogram is defined as the product of its base and its corresponding height (altitude). [Per the standard geometric theorem, a parallelogram can be transformed into a rectangle of equal base and height by translating a right-angled triangular section from one side to the other, thereby preserving the total area].
The formula is mathematically expressed as:
$A = b \times h$
Step 2: Algebraic Substitution and Manipulation
Substitute the known values into the area formula:
$48.72 = b \times 8.4$
To isolate the unknown variable $b$, apply the Division Property of Equality by dividing both sides of the equation by the height ($8.4$):
$b = \frac{48.72}{8.4}$
Step 3: Arithmetic Calculation
To simplify the division of decimal numbers, multiply both the numerator and the denominator by $10$ to shift the decimal point of the divisor, making it a whole number:
$b = \frac{48.72 \times 10}{8.4 \times 10}$
$b = \frac{487.2}{84}$
Performing the long division:
- $84 \times 5 = 420$
- $487.2 - 420 = 67.2$
- $84 \times 0.8 = 67.2$
Summing the quotients yields:
$b = 5.8 \text{ cm}$
Geometric Visualization
Below is a precisely scaled vector representation of the parallelogram. The dimensions are scaled by a factor of $30$ units per centimeter for visual clarity (Base $= 174$ units, Height $= 252$ units).
Final Solution: The missing value for the Base is $5.8 \text{ cm}$.
More Questions from Class 9 Mathematics Coordinate Geometry EXERCISE 9.1
- Q1(a): Find the area of each of the following parallelograms: (a)
- Q1(b): Find the area of each of the following parallelograms: (b)
- 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(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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