Question

LMNP is a rectangle. Find the value of x and the length of each diagonal.

LN = 7x + 11 and MP = 9x-5
X= ???

Answer 1

To find the value of x, we equate the lengths of the two sides of the rectangle and solve for x. Then, we use the Pythagorean theorem to find the length of each diagonal.

Step-by-step explanation:

To find the value of x, we can equate the lengths of the two sides of the rectangle: LN = MP. So, we have:

7x + 11 = 9x - 5

Subtracting 7x from both sides, we get:

11 = 2x - 5

Adding 5 to both sides, we get:

16 = 2x

Dividing both sides by 2, we get:

x = 8

Now, to find the length of each diagonal, we can use the Pythagorean theorem since the diagonals of a rectangle are its hypotenuses:

Let's call one diagonal KL and the other diagonal MN.

Using the lengths of LN and MP, we can construct the following system of equations:

KL^2 = LN^2 + MP^2

MN^2 = LN^2 + MP^2

Substituting the values, we get:

KL^2 = (7(8) + 11)^2 + (9(8) - 5)^2

MN^2 = (7(8) + 11)^2 + (9(8) - 5)^2

Simplifying and solving for each diagonal, we get:

KL ≈ 291.59

MN ≈ 291.59

Answer 2

Step-by-step explanation:

The diagonals of a rectangle are congruent, thus

MP = LN , substitute values

9x - 9 = 7x + 9 ( subtract 7x from both sides )

2x - 9 = 9 ( add 9 to both sides )

2x = 18 ( divide both sides by 2 )

x = 9

LN = 7x + 9 = (7 × 9) + 9 = 63 + 9 = 72

MP = 9x - 9 = (9 × 9) - 9 = 81 - 9 = 72