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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 $DL$.

Solution :

Initial Setup & Given Variables

Let us define the geometric properties and given parameters of the parallelogram $ABCD$:

  • Area of parallelogram $ABCD = 1470 \text{ cm}^2$
  • Length of base $AB = 35 \text{ cm}$
  • Length of base $AD = 49 \text{ cm}$
  • $DL \perp AB$, establishing $DL$ as the corresponding altitude (height) to the base $AB$.
  • $BM \perp AD$, establishing $BM$ as the corresponding altitude (height) to the base $AD$.

Geometric Visualization

The following high-precision diagram models the parallelogram to scale, where $10 \text{ units} = 1 \text{ cm}$. The exact coordinates are calculated using trigonometric projections to ensure spatial accuracy ($\angle DAB \approx 59^\circ$).

A B C D L M AB = 35 cm AD = 49 cm DL BM Area = 1470 cm²

Step 1: Calculating the Length of Altitude $DL$

The area of a parallelogram is defined by the product of any chosen base and its corresponding perpendicular altitude [Per the Euclidean geometric principle of area equivalence].

The fundamental formula is:

$\text{Area} = \text{Base} \times \text{Corresponding Height}$

Taking $AB$ as the base, the corresponding height is the perpendicular segment $DL$. Substituting the known values into the area equation:

$\text{Area}(ABCD) = AB \times DL$

$1470 = 35 \times DL$

Isolating $DL$ by dividing both sides by $35$:

$DL = \frac{1470}{35}$

$DL = 42 \text{ cm}$

Step 2: Calculating the Length of Altitude $BM$

The area of the parallelogram remains constant regardless of which base is chosen for the calculation [By the invariant property of 2D geometric areas]. We now apply the area formula using $AD$ as the base. The corresponding height for base $AD$ is the perpendicular segment $BM$.

$\text{Area}(ABCD) = AD \times BM$

Substituting the known area and the length of base $AD$:

$1470 = 49 \times BM$

Isolating $BM$ by dividing both sides by $49$:

$BM = \frac{1470}{49}$

$BM = 30 \text{ cm}$

Final Conclusion

Final Solution: The length of the altitude $DL$ is $42 \text{ cm}$ and the length of the altitude $BM$ is $30 \text{ cm}$.


More Questions from Class 9 Mathematics Coordinate Geometry EXERCISE 9.1


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