CameraIcon
CameraIcon
SearchIcon
MyQuestionIcon


Question

In the given Figure, diagonals $$AC$$ and $$BD$$ of quadrilateral $$ABCD$$ intersect at $$O$$ such that $$OB = OD$$. If $$AB = CD$$, then show that :
(i) $$ar(DOC) = ar(AOB)$$
(ii) $$ar(DCB) = ar(ACB)$$
(iii) $$DA\parallel CB$$ or $$ABCD$$ is a parallelogram.
463928_be8a25276a83417e9c57678a8495e320.png


Solution

Construction: 

Draw $$DE \bot AC$$ and $$BF \bot AC$$.

In $$\triangle DEO$$ and $$\triangle BFO$$

$$\angle DEO= \angle BFO$$          ....By construction 

$$\angle DOE= \angle BOF$$          ....Vertically opposite angles 
$$OD = OB$$                     .....Given

$$\triangle DOE \cong \triangle BOF$$     ...SAA test of congruence    ....(1)

$$\therefore DE=BF$$          .... CSCT        ....(2)

and 

$$OE = OF$$            ... CSCT  ....(3)


In $$\triangle DCE$$ and $$\triangle BFA$$ 

$$\angle DEC= \angle BFA$$          ..... By construction

$$DC = AB$$         ....Given
$$DE = BF$$          ....From (2)

$$\triangle DEC \cong \triangle BFA$$    .....(RHS) Right Hypotenuse Side test    ....(4)

$$\therefore EC = AF$$       ......C.S.C.T.      ....(5)

Similarly, we can prove $$\triangle DOA \cong \triangle BOC$$    ....(6)

Adding (1) and (2), we get 

$$OE+EC = OF+AF$$

$$\therefore OC = OA$$    .....(7)


(i) Adding (1) & (4), we get,

$$A(\triangle DOE) + A (\triangle DEC) = A (\triangle BOF) + A (\triangle BFA) $$

$$\therefore A(\triangle DOC) = A (\triangle BOA)$$      ....(8)


(ii) Adding (6) and (8), we get

$$A (\triangle DOC) + A (\triangle DOA) = A (\triangle BOA) + A (\triangle BOC) $$
$$\Rightarrow A (\triangle ACD) = A (\triangle ACB)$$


(iii) So, from (7) and given,

$$ \Box ABCD$$ is a parallelogram       .....(Diagonals bisect each other)

and $$DA \parallel CB$$      ....Opp. sides of parallelogram are parallel to each other


498089_463928_ans_2e5eb87811014203b9ba7312902d052e.png

Mathematics
RS Agarwal
Standard IX

Suggest Corrections
thumbs-up
 
0


similar_icon
Similar questions
View More


similar_icon
Same exercise questions
View More


similar_icon
People also searched for
View More



footer-image