A B C D is a parallelogram such that ∠ B A D = 120 ∘ and A C bisects ∠ B A D show that A B C D is a rhombus.

Given ∠ B A D = 120 ∘ and A C bisects ∠ B A D ∠ B A C = 1 2 × 120 ∘ = 60 ∘ ∠ 1 = ∠ 2 = 60 ∘ A D ‖ B C and A C is the traversal ∠ 2 = ∠ 4 = 60 ∘ ∆ A B C is isosceles triangle ∵ ∠ 1 = ∠ 4 = 60 ∘ ⇒ A B = B C Parallelogram A B C D is a rhombus.

Find the length of a chord which is at a distance of 2 11 cm from the centre of a circle of radius 12 cm .

Let A B be the chord and C be the mid point of A B Therefore, O C ⊥ A B Join O A and O C . O A is the radius Given O C = 2 11 cm and O A = 12 cm In a right ∆ O A C , using Pythagoras Theorem, we get, A C 2 = O A 2 - O C 2 = 12 2 - 2 11 2 = 144 - 44 = 100 cm A C 2 = 100 cm A C = 10 cm Therefore, length of the chord A B = 2 A C = 2 × 10 cm = 20 cm

Find the value of x ∘ in the following figures:

Using the theorem the angle subtended by an arc of a circle at the centre is double the angle subtended by it at any point on the remaining part of a circle. ∠ P O R = 2 ∠ P Q R x ∘ = 2 × 50 ∘ x ∘ = 100 ∘

Find the value of x ∘ in the following figures:

Using the theorem the angle subtended by an arc of a circle at the centre is double the angle subtended by it at any point on the remaining part of a circle. ∠ M N L = 1 2 Reflex ∠ M O L = 1 2 × 260 ∘ x ∘ = 130 ∘

Find the value of x ∘ in the following figures:

Using the theorem the angle subtended by an arc of a circle at the centre is double the angle subtended by it at any point on the remaining part of a circle. X Y is the diameter of the circle. Therefore ∠ X Z Y = 90 ∘ (Angle on a semi - circle) In ∆ X Y Z x ∘ + 63 ∘ + 90 ∘ = 180 ∘ x ∘ = 27 ∘

E M B I B E

Mathematics Standard 9>Chapter 4 - Geometry>Solved Example>Q 1

MEDIUM

9th Tamil Nadu Board

IMPORTANT

Earn 100

In a parallelogram $A B C D$ , the bisectors of the consecutive angles $∠ A$ and $∠ B$ insersect at $P$ . Show that $∠ A P B = 90^{\circ}$

Important Questions on Geometry

MEDIUM

9th Tamil Nadu Board

IMPORTANT

Mathematics Standard 9>Chapter 4 - Geometry>Solved Example>Q 2

In the following figure $A B C D$ is a parallelogram, $P$ and $Q$ are the midpoints of sides $A B$ and $D C$ respectively. Show that $A P C Q$ is a parallelogram.

Question Image

MEDIUM

9th Tamil Nadu Board

IMPORTANT

Mathematics Standard 9>Chapter 4 - Geometry>Solved Example>Q 3

A B C D

is a parallelogram such that

∠ B A D = 120^{\circ}

and

A C

bisects

∠ B A D

show that

A B C D

is a rhombus.

MEDIUM

9th Tamil Nadu Board

IMPORTANT

Mathematics Standard 9>Chapter 4 - Geometry>Solved Example>Q 4

In a parallelogram

A B C D, P

and

Q

are the points on line

D B

such that

P D = B Q

show that

A P C Q

is a parallelogram

MEDIUM

9th Tamil Nadu Board

IMPORTANT

Mathematics Standard 9>Chapter 4 - Geometry>Solved Example>Q 5

Find the length of a chord which is at a distance of $2 \sqrt{11} cm$ from the centre of a circle of radius $12 cm$ .

Question Image

HARD

9th Tamil Nadu Board

IMPORTANT

Mathematics Standard 9>Chapter 4 - Geometry>Solved Example>Q 6

In the concentric circles, chord $A B$ of the outer circle cuts the inner circle at $C$ and $D$ as shown in the diagram. Prove that, $A B - C D = 2 A C$