Question

in the figure triangle ADB and triangle cdb are on the same base DB if AC and BD intersect at O then prove that area of triangle ADB upon area of triangle CBD is equals AO upon CO​

Answer 1

Explanation:

Given that ΔADB and ΔCDB are on the same base DB.

Also, given that AC and BD intersect at O.

To prove that  $\frac{a r \Delta A DB }{a r \Delta CBD}=\frac{A O}{C O}$

Since, we know that the area of the triangle can be determined using the formula,

$\text { Area of triangle }=\frac{1}{2} \times \text { Base } \times \text { Height}$

Now, we shall draw $\mathrm{AE} \perp B D$ and $\mathrm{CF} \perp B D$

Thus, the area of the ΔADB is given by

$\text { ar } \Delta \mathrm{ADB}=\frac{1}{2} \times B D \times A E$

The area of ΔCDB is given by

$\text { ar } \Delta \mathrm{CBD}=\frac{1}{2} \times D B \times C F$

Dividing the two triangles, we have,

$\frac{a r \Delta A DB }{a r \Delta CBD}=\frac{\frac{1}{2} \times B D \times A E}{\frac{1}{2} \times D B \times C F}$

Cancelling the common terms, we get,

$\frac{a r \Delta A DB }{a r \Delta CBD}=\frac{ A E}{ C F}$  ----------- (1)

Now, let us consider the ΔAOE and ΔCOF

From the triangles, we can see that,

∠AEO = ∠CFO (right angles)

Also, we can see that,

∠AOE = ∠COF (vertically opposite angles)

Then, by AA Similarity Criterion, we have,

$\Delta \mathrm{AOE} \sim \Delta \mathrm{COF}$

Also, we know that if two trianlges are similar then their corresponding sides are in the same ratio.

Thus, we have,

$\frac{AE}{CF} =\frac{AO}{CO}$ -----------(2)

Substituting the equation (2) in equation (1), we have,

$\frac{a r \Delta A DB }{a r \Delta CBD}=\frac{A O}{C O}$

Hence proved

Learn more:

(1) In the figure given below, ABC and DBC are two triangles on the same base BC. if AD is intersect BC at O then show that :ar(ABC)/ar(DBC) = AO/DO

brainly.in/question/5825734

(2) Question 4 In the given figure, ABC and ABD are two triangles on the same base AB. If line-segment CD is bisected by AB at O, show that ar (ABC) = ar (ABD).

Class 9 - Math - Areas of Parallelograms and Triangles Page 162"

brainly.in/question/1427774

Answer 2

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