Question
Question: In the figure, ABCD is a quadrilateral. AC is the diagonal and \(DE\left\| {AC} \right.\) also DE me...
In the figure, ABCD is a quadrilateral. AC is the diagonal and DE∥AC also DE meets BC produced at E. Show that ar(ABCD)=ar(ΔABE).
Solution
According to given in the question we have to show that ar(ABCD)=ar(ΔABE) when ABCD is a quadrilateral. AC is the diagonal and DE∥AC also DE meets BC produced at E. So, first of all as mentioned in the question that AC is a diagonal and DE∥ACalso DE meets BC produced at E. therefore we can determine the two triangles which are in between the quadrilateral ABCD.
Now, we have to make both the triangles ΔDAC and ΔEAC which can be done by comparing the angles and sides of both of the triangles.
Now, we have to add the areas of both of the triangles ΔDAC and ΔEAC after that we have to add the area of the triangle ΔABC in the area of the triangles we just obtained.
Complete step by step answer:
Step 1: First of all we as mentioned in the solution hint, that AC is a diagonal and DE∥AC also DE meets BC produced at E. Therefore we can determine the two triangles which are in between the quadrilateral ABCD. Hence,
⇒ar(ABCD)=ar(ΔABC)+ar(ΔDAC)..............(1)
Step 2: Now, we have to Now, we have to make the both the triangles ΔDAC and ΔEAC which can be done by comparing the angles and sides of the both of the triangles as we know that ΔDAC and ΔEAC lies on the same base and line DE is parallel to the line AC. Hence,
⇒ar(ΔDAC)=ar(ΔEAC)................(2)
Step 3: Now, we have to add the area of the ΔABC in the expression (2) as obtained in the solution step 2 according to mentioned in the solution hint,
⇒ar(ΔDAC)+ar(ΔABC)=ar(ΔEAC)+ar(ΔABC)…………………..(3)
Step 4: Now, as we know that ΔDAC and ΔABC is a quadrilateral ABCD and ΔEAC and ΔABC is the triangle ΔABE. Hence, form the expression (3) as obtained in the solution step 3.
⇒ar(ABCD)=ar(ΔABE)
Hence, we have proved that ar(ABCD)=ar(ΔABE) when ABCD is a quadrilateral. AC is the diagonal and DE∥AC also DE meets BC produced at E.
Note: It is necessary to make both of the triangles ΔDAC and ΔEAC with the help of the AA rule in which we have to make to angles similar to each other to make the triangles congruent to each other to determine areas.
In the given quadrilateral ABCD there are two triangles which are ΔDAC and ΔABC so the sum of areas of both of the triangles will be equal to the area of the quadrilateral ABCD.