Question

given lnop is a rectangle, lo = 15x + 19, and qn = 10x + 2. find pn.
pn =

Explanation:

Step1: Recall rectangle diagonals property

In a rectangle, diagonals are equal and bisect each other. So, \( LO = PN \) and \( QN=\frac{1}{2}LO \) (since \( Q \) is the mid - point of the diagonals).

Step2: Set up the equation

Since \( QN=\frac{1}{2}LO \), we substitute the given expressions:
\( 10x + 2=\frac{1}{2}(15x + 19) \)
Multiply both sides by 2 to eliminate the fraction:
\( 2(10x + 2)=15x + 19 \)
\( 20x+4 = 15x + 19 \)

Step3: Solve for \( x \)

Subtract \( 15x \) from both sides:
\( 20x-15x+4=15x - 15x+19 \)
\( 5x+4 = 19 \)
Subtract 4 from both sides:
\( 5x+4 - 4=19 - 4 \)
\( 5x=15 \)
Divide both sides by 5:
\( x = 3 \)

Step4: Find \( LO \) (and thus \( PN \))

Substitute \( x = 3 \) into \( LO=15x + 19 \):
\( LO=15\times3+19=45 + 19=64 \)
Since \( LO = PN \) (diagonals of a rectangle are equal), \( PN = 64 \)

Answer:

\( 64 \)