In rectangle ABCD, the diagonals intersect at point O Find the perimeter of triangle AOC if A B is 4cm

In rectangle ABCD, the diagonals intersect at point O Find the perimeter of triangle AOC if A B is 4cm centimeters BD is 5 cm?

Let the diagonals AC and BD in rectangle ABCD intersect at point O, with side AB = 4 cm, diagonal BD = 5 cm.Then:

AO = OS = BO = OD = BD: 2 = 5: 2 = 2.5 (cm) – since the diagonals are equal and the intersection point is divided in half according to the property of the rectangle diagonals.

Consider a right-angled triangle ABD (∠BAD = 90º), in it, according to the Pythagorean theorem, BD² = AB² + AD², then:

AD² = BD² – AB² or

AD² = 5² – 4²;

AD = 3 cm.

The diagonals divide the rectangle into two pairs of equal triangles: ∆AOB = ∆СОD; ∆AOD = ∆СОВ. Let’s find the perimeters of these triangles as the sum of their sides:

P (∆AOB) = AO + OB + AB;

P (∆AOB) = 2.5 + 2.5 + 4 = 9 (cm);

P (∆AOD) = AO + OD + AD;

P (∆AOD) = 2.5 + 2.5 + 3 = 8 (cm).

Answer: The perimeters of the triangles are 8 cm and 9 cm.