Question

ABCD is a cyclic quadrilateral in which AB =14.4 cm,BC=12.8cm,and CD=9.6cm. If AC bisects BD, then what is the length of AD?

Answer 1

Length of AD is 19.2 cm

Step-by-step explanation:

Length of AD is 19.2 cm

ΔAPB and ΔDPC are similar and as per the similarity criteria-

q/a = 9.6/14.4 = a/p ''''''''(1)

Therefore, p = 3a/2 '"""""(2)

& q = 2a/3 """""(3)

Now, As per the Apollonian theorem -

The sum of squares of any of the two sides of a triangle equals to twice its square on half of the third side, along with the twice of its square on the median bisecting the third side.

Applying this theorem,

In ΔADB, $x^{2} + 14.4^{2} = 2(p^{2} + a^{2})$

$9.6^{2} + 12.8^{2} = 2(q^{2}+a^{2})\\9.6^{2} + 12.8^{2} = 2(4/9a^{2} + a^{2})\text {Using (3)}\\\text {Therefore} \\ a^{2} = \frac {[9.6^{2} + 12.8^{2}]}  {2\times13} \\So, a^{2} = 128\times9/13\\\text {Now, Using equation (2)}\\x^{2} + 14.4^{2} = 2(9/4a^{2} + a^{2})\\Hence, x^{2} = (2 \times 13 \times 128 \times 9)/13 \times 4 - 14.4^{2} = 368.64\\\text {Therefore, x = 19.2}$

Answer 2

ABCD is a cyclic quadrilateral in which AB = 14.4 cm, BC = 12.8 cm and CD = 9.6 cm.

If AC bisects BD, then what is the length of AD ?

Using the theorem,

“ If either of the diagonal s of a cyclic quadrilateral bisects the other diagonal, then the opposite side of the quadrilateral will be in the same ratio.”

• See question, here it said that AC (one diagonal of cyclic quadrilateral) bisected the other one BD.

so, the ratio of the opposite sides will be in the same ratio.

i.e., AB/CD = AD/BC

here, AB = 14.4 cm , BC = 12.8 cm and CD = 9.6 cm

⇒14.4/9.6 = AD/12.8

⇒3/2 = AD/12.8

⇒AD = 6.4 × 3 = 19.2 cm

Therefore the length of AD is 19.2 cm