Question

the diagram below shows the rectangle boar.
find the length of \\(\overline{pa}\\).
if entering your answer as a decimal, round your final answer to the nearest hundredth.
\\(pa = \square\\) units

Explanation:

Step1: Recall Rectangle Diagonal Property

In a rectangle, diagonals are equal and bisect each other. So, \( RO = BA \) (opposite sides of rectangle), and diagonals \( RA \) and \( BO \) bisect each other? Wait, no, in rectangle \( BOAR \), the diagonals are \( BR \) and \( OA \)? Wait, no, the rectangle is \( BOAR \), so vertices are \( B, O, A, R \) in order. So sides: \( BO \) and \( AR \) are opposite, \( BA \) and \( OR \) are opposite? Wait, maybe better to use the right triangle. Wait, \( BO = 12 \), \( BR = 5 \)? Wait, no, the diagram: \( B \) to \( R \) is 5, \( B \) to \( O \) is 12. Since it's a rectangle, \( RA = BO = 12 \)? No, wait, in rectangle \( BOAR \), the diagonals are \( RO \) and \( BA \)? Wait, maybe I misread. Wait, the key is that in a rectangle, diagonals are equal and the distance from the midpoint \( P \) (since \( P \) is on \( OA \) with right angle, maybe \( P \) is the midpoint? Wait, no, the dashed line is from \( P \) to \( A \), perpendicular? Wait, no, maybe \( OA \) is a diagonal, and \( P \) is the midpoint? Wait, no, let's think again.

Wait, \( BOAR \) is a rectangle, so \( BR \) and \( OA \) are diagonals? Wait, \( B \), \( O \), \( A \), \( R \): so \( BO \) is a side, \( OA \) is a side, \( AR \) is a side, \( RB \) is a side? No, rectangle has four right angles. So \( \angle B \), \( \angle O \), \( \angle A \), \( \angle R \) are right angles? Wait, the diagram shows \( B \) connected to \( R \) (length 5), \( B \) connected to \( O \) (length 12), \( O \) connected to \( A \), \( A \) connected to \( R \), and \( P \) is on \( OA \) with a right angle to \( A \). Wait, maybe \( RO \) is a diagonal, and \( PA \) is the distance from \( P \) to \( A \), where \( P \) is the midpoint? Wait, no, maybe we can use the Pythagorean theorem. Wait, in rectangle \( BOAR \), the length of \( RA \) can be found? Wait, no, \( BR = 5 \), \( BO = 12 \), so the diagonal \( OR \) (wait, \( B \) to \( R \) is 5, \( B \) to \( O \) is 12, so triangle \( BRO \) is right-angled? Wait, no, in rectangle, adjacent sides are perpendicular. So \( BR \) and \( BO \) are adjacent sides, so \( \angle B \) is right angle. So \( BR = 5 \), \( BO = 12 \), so the diagonal \( RO \) (from \( R \) to \( O \)) would be \( \sqrt{5^2 + 12^2} = 13 \). Then, in rectangle \( BOAR \), the diagonals \( RO \) and \( BA \) are equal, so \( BA = 13 \). Wait, but \( P \) is the midpoint of \( OA \)? Wait, no, the problem is to find \( PA \). Wait, maybe \( OA \) is equal to \( BR = 5 \)? No, this is confusing. Wait, maybe the rectangle has length 12 and width 5, so the diagonal is 13, and \( PA \) is half of the other diagonal? Wait, no. Wait, let's re-express:

In rectangle \( BOAR \), sides: \( BO = 12 \), \( BR = 5 \) (since \( B \) to \( R \) is 5, and \( \angle B \) is right angle). So the diagonal \( OR \) (from \( O \) to \( R \)) is \( \sqrt{5^2 + 12^2} = 13 \). Now, in a rectangle, diagonals are equal, so \( BA = OR = 13 \)? No, \( BA \) is a side? Wait, no, the rectangle's vertices are \( B, O, A, R \), so the sides are \( BO \), \( OA \), \( AR \), \( RB \). So \( BO \) and \( AR \) are opposite, \( OA \) and \( RB \) are opposite. So \( BO = AR = 12 \), \( RB = OA = 5 \)? No, that can't be, because then the diagonal would be 12, but we have a right triangle with legs 5 and 12. Wait, I think I mixed up the sides. Let's assume that \( BR = 5 \) and \( BO = 12 \) are adjacent sides (perpendicular), so the diagonal \( RO \) is 13 (by Pythagoras: \( 5^2 + 12^2 = 25 + 144 = 169 = 13^2 \)). Now, in rectangle \( BOAR \), the diagonals are \( RO \) and \( BA \), which are equal (13). Now, \( P \) is the midpoint of \( OA \)? Wait, no, the problem is to find \( PA \). Wait, maybe \( OA \) is equal to \( BR = 5 \), and \( PA \) is the distance from \( P \) to \( A \), where \( P \) is the midpoint? No, this is not working. Wait, maybe the rectangle has length 12 and width 5, so the diagonal is 13, and \( PA \) is half of the diagonal? Wait, no, 13/2 = 6.5? But that's 6.5, which is 6.50. Wait, let's check:

If \( BOAR \) is a rectangle with length 12 and width 5, then the diagonals are 13. The midpoint of the diagonal \( OA \) (wait, no, the diagonals are \( BR \) and \( OA \)). Wait, \( BR \) is the diagonal with length 13, and \( OA \) is the other diagonal, also length 13. Then \( P \) is the midpoint of \( OA \), so \( PA = OA/2 = 13/2 = 6.5 \). Ah, that makes sense! Because in a rectangle, diagonals bisect each other, so the midpoint \( P \) divides \( OA \) into two equal parts. So \( OA = BR = \sqrt{5^2 + 12^2} = 13 \), so \( PA = 13/2 = 6.5 \).

Step1: Calculate Diagonal Length

In rectangle \( BOAR \), \( BR = 5 \) and \( BO = 12 \) are perpendicular sides. Using Pythagorean theorem, diagonal \( OA = BR = \sqrt{5^2 + 12^2} \).
\( \sqrt{25 + 144} = \sqrt{169} = 13 \).

Step2: Find \( PA \) (Midpoint Property)

In a rectangle, diagonals bisect each other. So \( P \) (midpoint of \( OA \)) divides \( OA \) into two equal segments. Thus, \( PA = \frac{OA}{2} \).
\( PA = \frac{13}{2} = 6.5 \).

Answer:

6.50