ABCD is a rectangle . if AM = 2y+5 and DM= 4y-15 . find the length of each diagonal. Related: Extra Questions and Practice Exercise - Understanding Quadrilaterals?

Question

ABCD is a rectangle . if AM = 2y+5 and DM= 4y-15 . find the length of each diagonal. Related:  Extra Questions and Practice Exercise - Understanding Quadrilaterals?

Answer

Answer

Solution to Problem:

Given:

ABCD is a rectangle.

AM = 2y + 5

DM = 4y - 15

To find:

The length of each diagonal of the rectangle.

Steps to solve the problem:

We know that in a rectangle, opposite sides are equal in length.

Let's assume the length of AB and CD as 'a' and the length of BC and DA as 'b'.

Therefore, we have a = b.

We also know that diagonals of a rectangle bisect each other.

Let's assume the length of AC as 'd' and the length of BD as 'e'.

Therefore, we have d = 2b and e = 2a.

Now, let's use the given information to form two equations.

From triangle AMD, we have AM^2 + DM^2 = AD^2.

Substituting the values of AM and DM, we get (2y + 5)^2 + (4y - 15)^2 = d^2.

On simplifying, we get 20y^2 - 16y + 200 = d^2.

From triangle BMC, we have BM^2 + MC^2 = BC^2.

Substituting the values of BM and MC, we get (2y - 5)^2 + (4y + 15)^2 = e^2.

On simplifying, we get 20y^2 + 16y + 200 = e^2.

Now, substituting the values of d and e in terms of a and b, we get:

4b^2 + 20a^2 - 16b^2 + 200 = d^2

4a^2 + 20b^2 + 16a^2 + 200 = e^2

On simplifying, we get:

24a^2 + 4b^2 + 200 = d^2

24b^2 + 4a^2 + 200 = e^2

Substituting a = b, we get:

28a^2 + 200 = d^2

28a^2 + 200 = e^2

On simplifying, we get:

a = b = 5

Therefore, d = 10 and e = 10.

Hence, the length of each diagonal of the rectangle is 10 units.

Answer:

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