Question

In ΔPQR, ∠P = 90°, PQ = 8 cm, QR = 17 cm. From midpoint M perpendiculars are drawn parallel to sides PQ and PR. Perimeter of rectangle PAMB is ______.

Options

• 23 cm

• 25 cm

• 24 cm

• 26 cm

Answer 1

In ΔPQR, ∠P = 90°, PQ = 8 cm, QR = 17 cm. From midpoint M perpendiculars are drawn parallel to sides PQ and PR. Perimeter of rectangle PAMB is 23 cm.

Explanation:

Given:

• ΔPQR with ∠P = 90°
• PQ = 8 cm
• QR = 17 cm
• M is the midpoint of QR
• Perpendiculars are drawn from M parallel to sides PQ and PR, forming rectangle PAMB.

We need to find the perimeter of rectangle PAMB.

Step 1: Find the length of PR

Using the Pythagorean theorem in ΔPQR:

`PR = sqrt(QR^2 - PQ^2)`

= `sqrt(17^2 - 8^2)`

= `sqrt(289 - 64)`

= `sqrt(225)`

= 15 cm

Step 2: Coordinates setup (optional for clarity)

Let’s place the triangle on the coordinate plane:

• P = (0, 0)
• Q = (8, 0) (since PQ = 8)
• R = (0, 15) (since PR = 15)

Step 3: Find the midpoint M of QR

Coordinates of Q = (8, 0) and R = (0, 15), so midpoint M is

`M = ((8 + 0)/2, (0 + 15)/2) = (4, 7.5)`

Step 4: Find points A and B

• From M, draw perpendiculars parallel to PQ and PR.
• PQ is along the x-axis (horizontal), so the line parallel to PQ through M is horizontal y = 7.5.
• PR is along the y-axis (vertical), so the line parallel to PR through M is vertical x = 4. 
• A is on PR (the y-axis), so A = (0, 7.5). 
• B is on PQ (the x-axis), so B = (4, 0).

Step 5: Calculate the lengths of sides of rectangle PAMB

• PA = distance between P(0, 0) and A(0, 7.5) = 7.5 cm
• PB = distance between P(0, 0) and B(4, 0) = 4 cm

Step 6: Calculate perimeter of PAMB

The perimeter of rectangle PAMB is 

2 × (PA + PB)

= 2 × (7.5 + 4)

= 2 × 11.5

= 23 cm