# the diagonals of a quadrilateral abcd intersect each other at the point o such that

### Mohammed

Guys, does anyone know the answer?

get the diagonals of a quadrilateral abcd intersect each other at the point o such that from screen.

## The diagonals of a quadrilateral ABCD intersect each other at the point O such that AOBO = CODO . Show that ABCD is a trapezium.

Click here👆to get an answer to your question ✍️ The diagonals of a quadrilateral ABCD intersect each other at the point O such that AOBO = CODO . Show that ABCD is a trapezium.

The diagonals of a quadrilateral ABCD intersect each other at the point O such thatQuestion BO AO = DO CO

. Show that ABCD is a trapezium.

Medium Open in App

Updated on : 2022-09-05

Given:Solution Verified by Toppr

The diagonals of a quadrilateral ABCD intersect each other at the point O such that

BO AO = DO CO i.e., CO AO = DO BO

To Prove: ABCD is a trapezium

Construction:

Draw OE∥DC such that E lies on BC.

Proof: In △BDC,

By Basic Proportionality Theorem,

OD BO = EC BE ............(1) But, CO AO = DO BO

(Given) .........(2)

∴ From (1) and (2) CO AO = EC BE

Hence, By Converse of Basic Proportionality Theorem,

OE∥AB Now Since, AB∥OE∥DC ∴ AB∥DC

Hence, ABCD is a trapezium.

Video Explanation

Solve any question of Triangles with:-

Patterns of problems

>

Was this answer helpful?

372 19

## The diagonals of a quadrilateral ABCD intersect each other at the point O such that AO/BO=CO/DO . Show that ABCD is a trapezium.

The diagonals of a quadrilateral ABCD intersect each other at the point O such that AO/BO=CO/DO . Show that ABCD is a trapezium.

Byju's Answer Standard X Mathematics

Basic Proportionality Theorem

The diagonals... Question

The diagonals of a quadrilateral

ABCD

intersect each other at the point

O such that AOBO=CODO . Show that ABCD is a trapezium. Open in App Solution

**Step 1. Explaining the diagram.**

Let ABCD

be quadrilateral where

AC and BD

intersects each other at

O such that, AOBO=CODO

**Step 2. Showing**

ABCD

**is trapezium**

**Construction-**From the point

O , draw a line EO touching AD at E in such a way that, EO∥DC∥AB In ΔDAB, EO || AB

By using** Basic Proportionality Theorem**

DEEA=DOOB........................(i)

Also, given, AOBO=CODO

⇒AOCO=BODO [applying alternendo]

⇒COAO=DOOB [ [applying invertendo]

⇒DOOB =COAO..........................(ii)

From equation (i) and (ii), We have DEEA=COAO

Therefore, By applying converse of **Basic Proportionality Theorem,**

EO || DC Also

EO || AB ⇒ AB || DC.

**Hence, quadrilateral**

ABCD

**is a trapezium with**

AB || CD.

Suggest Corrections 16

SIMILAR QUESTIONS

**Q.**

ABCD is a trapezium in which AB || DC and its diagonals intersect each other at the point O. Show that

**Q.**The diagonals of a quadrilateral

A B C D

intersects each other at the point

O such that A O B O = C O D O . Show that A B C D is a trapezium.

**Q.**

**Question 10**

The diagonals of a quadrilateral ABCD intersect each other at the point O such that

A O B O = C O D O

. Show that ABCD is a trapezium.

**Q.**The diagonals of a quadrilateral

A B C D

intersect each other at the point

O , such that A O B O = C O D O . Then A B C D is a trapezium.

**Q.**

A B C D

is a trapezium in which

A B | | D C

and its diagonals intersect each other at point

′ O ′ . Show that A O B O = C O D O . View More RELATED VIDEOS

Basic Proportionality Theorem

MATHEMATICS Watch in App EXPLORE MORE

Basic Proportionality Theorem

Standard X Mathematics

## Ex 6.2, 10

Ex 6.2, 10 The diagonals of a quadrilateral ABCD intersect each other at the point O such that 𝐴𝑂/𝐵𝑂 = 𝐶𝑂/𝐷𝑂 . Show that ABCD is a trapezium Given: ABCD is a quadrilateral where diagonals AC & BD intersect at O & 𝐴𝑂/𝐵𝑂=𝐶𝑂/𝐷𝑂 To prove: ABCD is a trapezium Construction: Let us draw a lin

**Check sibling questions**

## Ex 6.2, 10 - Chapter 6 Class 10 Triangles (Term 1)

Last updated at March 16, 2023 by Teachoo

Ads by

Ad

This video is only available for Teachoo black users

Subscribe Now

Get live Maths 1-on-1 Classs - Class 6 to 12

Book 30 minute class for ₹ 499 ₹ 299

### Transcript

Ex 6.2, 10 The diagonals of a quadrilateral ABCD intersect each other at the point O such that 𝐴𝑂/𝐵𝑂 = 𝐶𝑂/𝐷𝑂 . Show that ABCD is a trapezium Given: ABCD is a quadrilateral where diagonals AC & BD intersect at O & 𝐴𝑂/𝐵𝑂=𝐶𝑂/𝐷𝑂 To prove: ABCD is a trapezium Construction: Let us draw a line EF II AB passing through point O. Proof: Given 𝐴𝑂/𝐵𝑂=𝐶𝑂/𝐷𝑂 ⇒ 𝐴𝑂/𝐶𝑂=𝐵𝑂/𝐷𝑂 Now, in ∆ 𝐴𝐷𝐵 EO II AB 𝐴𝐸/𝐷𝐸=𝐵𝑂/𝐷𝑂 ⇒ 𝐴𝐸/𝐷𝐸=𝐴𝑂/𝐶𝑂 Thus in Δ ADC, Line EO divides the triangle in the same ratio ∴ EO II DC Now, EO II DC But, we know that EO II AB ⇒ EO II AB II DC ⇒ AB II DC Hence, one pair of opposite sides of quadrilateral ABCD are parallel Therefore ABCD is a trapezium . Hence proved

**Next**: Ex 6.3 →

Ask a doubt Facebook Whatsapp

Made by

### Davneet Singh

Davneet Singh has done his B.Tech from Indian Institute of Technology, Kanpur. He has been teaching from the past 13 years. He provides courses for Maths, Science, Social Science, Physics, Chemistry, Computer Science at Teachoo.

Guys, does anyone know the answer?