Ar(||gm)=2×ar.TriangleABC... Now.. Perpendicular distance =(2×area||gm)/Base...

Given:
Parallelogram ABCD
AB = 6cm
BC = 5cm
AC = 7cm

To Find:
Perpendicular distance between AB and CD

Solution:
To find the perpendicular distance between AB and CD, we need to understand the properties of a parallelogram.

Properties of a Parallelogram:
1. Opposite sides of a parallelogram are equal in length.
2. Opposite angles of a parallelogram are equal.
3. The diagonals of a parallelogram bisect each other.
4. The diagonals of a parallelogram divide it into four congruent triangles.

From the given information, we can determine that opposite sides AB and CD of the parallelogram ABCD are equal in length. Therefore, CD = 6 cm.

Using the Law of Cosines:
We can use the Law of Cosines to find the length of the diagonal BD.

The Law of Cosines states that in any triangle ABC, the square of one side is equal to the sum of the squares of the other two sides minus twice the product of the two sides multiplied by the cosine of the included angle.

In triangle ABC:
AC² = AB² + BC² - 2 * AB * BC * cos(∠ABC)

Plugging in the given values:
7² = 6² + 5² - 2 * 6 * 5 * cos(∠ABC)

Simplifying the equation:
49 = 36 + 25 - 60 * cos(∠ABC)
49 = 61 - 60 * cos(∠ABC)

Rearranging the equation:
cos(∠ABC) = (61 - 49) / 60
cos(∠ABC) = 12 / 60
cos(∠ABC) = 0.2

Now, we can find the measure of ∠ABC by taking the inverse cosine (cos⁻¹) of 0.2 using a scientific calculator or trigonometric tables. Let's assume ∠ABC = θ.

∠ABC = cos⁻¹(0.2)
∠ABC ≈ 78.46°

Using Trigonometry:
To find the perpendicular distance between AB and CD, we can use trigonometry.

In triangle ABC, we have the following information:
AC = 7 cm (opposite side)
BC = 5 cm (adjacent side)
∠ABC = θ ≈ 78.46° (included angle)

Using the trigonometric ratio:
tan(∠ABC) = AC / BC

Plugging in the values:
tan(78.46°) = 7 / 5

Simplifying the equation:
tan(78.46°) ≈ 1.4

Now, we can find the perpendicular distance between AB and CD using the formula:
Perpendicular distance = AB * tan(∠ABC)

Plugging in the values:
Perpendicular distance = 6 * 1.4
Perpendicular distance ≈ 8.4 cm

Therefore, the perpendicular distance between AB and CD in parallelogram ABCD is approximately 8.4 cm.