Answer

Verified

448.2k+ views

Hint: A trapezium is a 2D shape which falls under the category of quadrilaterals. A trapezium has two parallel sides and two non-parallel sides. Using this information first draw the diagram with the given data and solve the problem accordingly.

Complete step-by-step answer:

In the given trapezium ABCD, \[\overline {AB} \parallel \overline {CD} \], \[M\] and \[N\]are the points on the traversals \[\overleftrightarrow {AD}\] and \[\overleftrightarrow {BC}\] respectively as shown in the below diagram.

Also, given \[\dfrac{{AM}}{{MD}} = \dfrac{{BN}}{{NC}} = \dfrac{2}{3}\]

So, clearly from the diagram, \[\overline {MN} \parallel \overline {AB} \] and \[\overline {AB} \parallel \overline {CD} \].

Given that diagonal \[\overline {AC} \] intersects \[\overline {MN} \] at \[O\].

In \[\Delta ADC\], \[\overline {MO} \parallel \overline {DC} \] such that \[M \in \overline {AD} {\text{ }}\& {\text{ }}O \in \overline {AC} \].

We know that by the Triangle Proportionality theorem, if a line parallel to one side of a triangle intersects the other two sides of the triangle, then the lines divide these two sides proportionally.

By using this property in \[\Delta ADC\], we have

\[ \Rightarrow \dfrac{{AM}}{{MD}} = \dfrac{{AO}}{{OC}}\]

But we have \[\dfrac{{AM}}{{MD}} = \dfrac{2}{3}\]

So, \[\dfrac{{AO}}{{OC}} = \dfrac{2}{3}\]

We have to find \[\dfrac{{AO}}{{AC}}\]. From the diagram, \[OC = AO + OC\]

\[

\Rightarrow \dfrac{{AO}}{{OC}} = \dfrac{{AO}}{{AO + OC}} \\

\Rightarrow \dfrac{{AO}}{{OC}} = \dfrac{2}{{2 + 3}} \\

\therefore \dfrac{{AO}}{{OC}} = \dfrac{2}{5} \\

\]

Thus, \[\dfrac{{AO}}{{OC}} = \dfrac{2}{5}\].

Note: The length of the mid-segment is equal to half of the sum of parallel bases in a trapezium. In this problem we have used both the properties of triangles as well as the properties of trapezium.

Complete step-by-step answer:

In the given trapezium ABCD, \[\overline {AB} \parallel \overline {CD} \], \[M\] and \[N\]are the points on the traversals \[\overleftrightarrow {AD}\] and \[\overleftrightarrow {BC}\] respectively as shown in the below diagram.

Also, given \[\dfrac{{AM}}{{MD}} = \dfrac{{BN}}{{NC}} = \dfrac{2}{3}\]

So, clearly from the diagram, \[\overline {MN} \parallel \overline {AB} \] and \[\overline {AB} \parallel \overline {CD} \].

Given that diagonal \[\overline {AC} \] intersects \[\overline {MN} \] at \[O\].

In \[\Delta ADC\], \[\overline {MO} \parallel \overline {DC} \] such that \[M \in \overline {AD} {\text{ }}\& {\text{ }}O \in \overline {AC} \].

We know that by the Triangle Proportionality theorem, if a line parallel to one side of a triangle intersects the other two sides of the triangle, then the lines divide these two sides proportionally.

By using this property in \[\Delta ADC\], we have

\[ \Rightarrow \dfrac{{AM}}{{MD}} = \dfrac{{AO}}{{OC}}\]

But we have \[\dfrac{{AM}}{{MD}} = \dfrac{2}{3}\]

So, \[\dfrac{{AO}}{{OC}} = \dfrac{2}{3}\]

We have to find \[\dfrac{{AO}}{{AC}}\]. From the diagram, \[OC = AO + OC\]

\[

\Rightarrow \dfrac{{AO}}{{OC}} = \dfrac{{AO}}{{AO + OC}} \\

\Rightarrow \dfrac{{AO}}{{OC}} = \dfrac{2}{{2 + 3}} \\

\therefore \dfrac{{AO}}{{OC}} = \dfrac{2}{5} \\

\]

Thus, \[\dfrac{{AO}}{{OC}} = \dfrac{2}{5}\].

Note: The length of the mid-segment is equal to half of the sum of parallel bases in a trapezium. In this problem we have used both the properties of triangles as well as the properties of trapezium.

Recently Updated Pages

How many sigma and pi bonds are present in HCequiv class 11 chemistry CBSE

Why Are Noble Gases NonReactive class 11 chemistry CBSE

Let X and Y be the sets of all positive divisors of class 11 maths CBSE

Let x and y be 2 real numbers which satisfy the equations class 11 maths CBSE

Let x 4log 2sqrt 9k 1 + 7 and y dfrac132log 2sqrt5 class 11 maths CBSE

Let x22ax+b20 and x22bx+a20 be two equations Then the class 11 maths CBSE

Trending doubts

Which are the Top 10 Largest Countries of the World?

The 3 + 3 times 3 3 + 3 What is the right answer and class 8 maths CBSE

Fill the blanks with the suitable prepositions 1 The class 9 english CBSE

How many crores make 10 million class 7 maths CBSE

Difference Between Plant Cell and Animal Cell

Difference between Prokaryotic cell and Eukaryotic class 11 biology CBSE

Give 10 examples for herbs , shrubs , climbers , creepers

Change the following sentences into negative and interrogative class 10 english CBSE

Write a letter to the principal requesting him to grant class 10 english CBSE