Question

Assertion: If square ABCD and ΔPQR have same perimeter then QR = 8 cm.

Reason: Perimeter = 28 cm, x = 11.

Options

• Both A and R are true and R is the correct reason for A.

• Both A and R are true but R is the incorrect reason for A.

• A is true but R is false.

• A is false but R is true.

Answer 1

Both A and R are true and R is the correct reason for A.

Explanation:

Given:

• Square ABCD and triangle PQR have the same perimeter.
• Perimeter = 28 cm.
• x = 11.

Step 1: Perimeter of square ABCD

Side of square ABCD

= (x – 4) cm 

= (11 – 4) cm

= 7 cm

Perimeter of square

= 4 × 7 

= 28 cm

So, the square’s perimeter is 28 cm.

Step 2: Perimeter of triangle PQR

Triangle PQR has sides PQ, QR, PR

Given: PQ = QR   ...(Marked equal)

QR = 8 cm   ...(From assertion)

PQ = 8 cm

PR = (x – 1) cm

= 11 – 1

= 10 cm

Perimeter of triangle = PQ + QR + PR

= 8 + 8 + 10

= 26 cm

At first glance, this indicates the triangle’s perimeter is 26 cm, which is not equal to 28 cm square’s perimeter.

Step 3: Reevaluating the condition of same perimeter

The assertion states “If square ABCD and ΔPQR have the same perimeter”, which means:

Perimeter of square ABCD = Perimeter of ΔPQR.

We have the square’s perimeter as 28 cm.

Let’s check if this perimeter equality implies QR = 8 cm.

Step 4: Express triangle’s perimeter in terms of QR

Since PQ = QR   ...(Isosceles triangle),

Let QR = PQ = y cm,

PR = (x – 1) = 10 cm with x = 11.

Triangle perimeter = PQ + QR + PR

= y + y + 10

= 2y + 10

Set this equal to the square’s perimeter:

2y + 10 = 28

2y = 18

y = 9 cm

For the triangle to have the same perimeter as the square 28 cm, QR must be 9 cm, not 8 cm.

But the assertion says QR = 8 cm if they have the same perimeter, which contradicts the calculation.