If A B C D is a cyclic quadrilateral, then prove that A C · B D = A B · C D + B C · A D .

Let, A B = a , B C = b , C D = c & D A = d . Using cosine rule in △ A B C & △ A D C we get : A C 2 = a 2 + b 2 - 2 a b cos B & A C 2 = c 2 + d 2 - 2 c d cos D Now, B + D = π . ⇒ cos B + cos D = 0 ⇒ A C 2 c d + a b = a 2 + b 2 c d + c 2 + d 2 a b ⇒ A C 2 = a 2 c d + c 2 a b b 2 c d + d 2 a b + a b Similarly, taking diagonal BD we get, B D 2 = a b + c d b d + c a d a + b c Multiplying both equations : ⇒ A D . B D 2 = a c + b d 2 ⇒ A D . B D = a c + b d ⇒ A D . B D = A B . C D + B C . A D . Hence, proved.

In a triangle A B C , the median to the side B C is of length 1 11 - 6 3 and it divides the angle A into angles of 30 ° and 45 ° . Find the length of the side B C .

Let AD be the median to side BC , which divides ∠ A in angle of 30 o & 45 o Also, Let BC = x ∴ BD = CD = x 2 Let ∠ ADC be θ . ∴ x 2 + x 2 cot θ = x 2 cot 30 o - x 2 cot 45 o ⇒ cot θ = 3 - 1 2 ∴ cos θ = 3 - 1 8 - 2 3 , sin θ = 2 8 - 2 3 Applying sine-rule in ∆ ADC , we get AD sin π - θ - 45 o = DC sin 45 o ⇒ AD sin θ + 45 o = x 2 1 2 ⇒ AD = x 2 sin 45 o · cos θ + cos 45 o sin θ ⇒ AD = x 2 cos θ + sin θ ⇒ 1 11 - 6 3 = x 2 3 - 1 8 - 2 3 + 2 8 - 2 3 ⇒ x = 2 8 - 2 3 11 - 6 3 3 + 1 ⇒ x = 2 8 - 2 3 3 + 1 2 11 - 6 3 ⇒ x = 2 8 - 2 3 44 - 24 3 + 22 3 - 36 ⇒ x = 2 8 - 2 3 8 - 2 3 = 2

E M B I B E

TRIGONOMETRY FOR JEE MAIN AND ADVANCED>Chapter 11 - Properties of Triangles>EXERCISE>Q 65

HARD

Mathematics

IMPORTANT

Earn 100

Given that the area of a polygon of $n$ sides circumscribed about a circle is to the area of the circumscribed polygon of $2 n$ sides as $3 : 2$ , find $n$ .

Important Questions on Properties of Triangles

MEDIUM

Mathematics

IMPORTANT

TRIGONOMETRY FOR JEE MAIN AND ADVANCED>Chapter 11 - Properties of Triangles>EXERCISE>Q 66

The area of regular polygon of

n

sides inscribed in a circle is to that of the same number of sides circumscribing the same circle as

3 : 4

. Find the value of

n

.

HARD

Mathematics

IMPORTANT

TRIGONOMETRY FOR JEE MAIN AND ADVANCED>Chapter 11 - Properties of Triangles>EXERCISE>Q 67

There are two regular polygons, the number of sides in one being the double the number in the other, and an angle of one polygon is to an angle of the other is