Question
In the given figure, $\triangle\text{ABC}$ and $\triangle\text{DBC}$ have the same base BC. If AD and BC intersect at O, prove that $\frac{\text{ar}(\triangle\text{ABC})}{\text{ar}(\triangle\text{DBC})}=\frac{\text{AO}}{\text{DO}}.$
✓
Answer
Construction: Draw $\text{AX}\perp\text{CO}$ and $\text{DY}\perp\text{BO}.$
As,
$\frac{\text{ar}(\triangle\text{ABC})}{\text{ar}(\triangle\text{DBC})}=\frac{\frac{1}{2}\times\text{AX}\times\text{BC}}{\frac{1}{2}\times\text{DY}\times\text{BC}}$
$\Rightarrow\frac{\text{ar}(\triangle\text{ABC})}{\text{ar}(\triangle\text{DBC})}=\frac{\text{AX}}{\text{DY}}\dots(\text{i})$
In $\triangle\text{ABC}$ and $\triangle\text{DBC},\angle\text{AXY}=\angle\text{DYO}=90^\circ(\text{By construction})$
$\angle\text{AOX}=\angle\text{DOY}(\text{Vertically opposite angles})$$\therefore\triangle\text{AXY}\sim\triangle\text{DYO }(\text{By AA criterion})$
This completes the proof.
Need a full question paper?
Generate a complete, print-ready paper with questions like this in minutes — across 16+ boards, with answer keys.
Explore more
Similar questions
In the middle of a rectangular field measuring $30\ m \times 20\ m$, a well of $7\ m$ diameter and $10\ m$ depth is dug. The earth so removed is evenly spread over the remaining part of the field. Find the height through which the level of the field is raised.
→On a horizontal plane there is a vertical tower with a flagpole on the top of the tower. At a point, 9 metres away from the foot of the tower, the angle of elevation of the top and bottom the flagpole are 60° and 30° respectively. Find the height of the tower and the flagpole mounted on it. $\big[\text{Take}\sqrt{3}=1.732\big]$
→Find the area of a parallelogram ABCD if three of its vertices are A(2, 4), $\text{B}(2+\sqrt{3},\ 5)$ and C(2, 6).
→Draw a pie diagram to represent the world population given in the following table after determining the value of a.
| Country | India | China | Russia | USA | Others | Total |
| Percentage poulation | 15 | 20 | a | a | 25 | 100 |
Find a natural number whose square diminished by $84$ is equal to thrice of $8$ more than the given number.
→A road which is 7m wide surrounds a circular park whose circumference is 352m. Find the area of the road.
A courier company delivered a parcel from Mumbai to Pune. The customer paid ₹ 531 to the courier company. Now, tax invoice shows ₹ 450 as taxable price, CGST is ₹ 40.50 and SGST is ₹ 40.50 , then find the rate of GST applicable in this transaction.
Solve for $x$ and $y$:
$6(ax + by) = 3a + 2b,$
$6(bx - ay) = 3b - 2a$
→$6(ax + by) = 3a + 2b,$
$6(bx - ay) = 3b - 2a$
Solve the following systems of equations:
$\frac{5}{\text{x}-1}+\frac{1}{\text{y}-2}=2,$
$\frac{6}{\text{x}-1}-\frac{3}{\text{y}-2}=1.$
→$\frac{5}{\text{x}-1}+\frac{1}{\text{y}-2}=2,$
$\frac{6}{\text{x}-1}-\frac{3}{\text{y}-2}=1.$
Solve the following system of equations by the method of cross-multiplication:
→$ax + by = a^2,$
$bx + ay = b^2.$