Question
In a $\triangle\text{ABC,D}\ \text{and E}$ are points on the sides AB and AC respectively such that DE || BC.If AD = 4x - 3, AE = 8x - 7, BD = 3x - 1 and CE = 5x - 3, find the value of x.
✓
Answer
Given: AD = 4x - 3, AE = 8x - 7, BD = 3x - 1 and CE = 5x - 3
by thales theorem
$\frac{\text{AD}}{\text{BD}}=\frac{\text{AE}}{\text{CE}}$
$\Rightarrow\frac{4\text{x}-3}{3\text{x}-1}=\frac{8\text{x}-7}{5\text{x}-3}$
$\Rightarrow(4\text{x}-3)(5\text{x}-3)=(8\text{x}-7)(3\text{x}-1)$
$\Rightarrow20\text{x}^2+9-15\text{x}-12\text{x}=24\text{x}^2-21\text{x}-8\text{x}+7$
$\Rightarrow9-7-27\text{x}+29\text{x}=24\text{x}^2-20\text{x}^2$
$\Rightarrow2+2\text{x}=4\text{x }^2$
$\Rightarrow4\text{x}^2-2\text{x }-2=0$
$\Rightarrow2\text{x}^2-2\text{x}+\text{x}-1=0$
$\Rightarrow2\text{x}(\text{x}-1)+(\text{x}-1)=0$
$\Rightarrow(2\text{x}+1)(\text{x}-1)=0$
$\Rightarrow2\text{x}+1=0\ \text{or}\ \text{x}-1=0$
$\Rightarrow2\text{x}=-1\ \text{or x}=1$
$\Rightarrow\text{x}=\frac{-1}{2}$
$\text{Thus, x}=\frac{-1}{2}\ \text{and x}=1$
by thales theorem
$\frac{\text{AD}}{\text{BD}}=\frac{\text{AE}}{\text{CE}}$
$\Rightarrow\frac{4\text{x}-3}{3\text{x}-1}=\frac{8\text{x}-7}{5\text{x}-3}$
$\Rightarrow(4\text{x}-3)(5\text{x}-3)=(8\text{x}-7)(3\text{x}-1)$
$\Rightarrow20\text{x}^2+9-15\text{x}-12\text{x}=24\text{x}^2-21\text{x}-8\text{x}+7$
$\Rightarrow9-7-27\text{x}+29\text{x}=24\text{x}^2-20\text{x}^2$
$\Rightarrow2+2\text{x}=4\text{x }^2$
$\Rightarrow4\text{x}^2-2\text{x }-2=0$
$\Rightarrow2\text{x}^2-2\text{x}+\text{x}-1=0$
$\Rightarrow2\text{x}(\text{x}-1)+(\text{x}-1)=0$
$\Rightarrow(2\text{x}+1)(\text{x}-1)=0$
$\Rightarrow2\text{x}+1=0\ \text{or}\ \text{x}-1=0$
$\Rightarrow2\text{x}=-1\ \text{or x}=1$
$\Rightarrow\text{x}=\frac{-1}{2}$
$\text{Thus, x}=\frac{-1}{2}\ \text{and x}=1$
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
$500$ persons have to dip in a rectangular tank which is $80\ m$ long and $50\ m$ broad. What is the rise in the level of water in the tank, if the average displacement of water by a person is $0.04m^3?$
→If $\angle1=\angle2$ and $\triangle\text{NSQ}\cong\triangle\text{MTR},$ then prove that $\triangle\text{PTS}\sim\triangle\text{PRQ}.$
→As observed from the top of a $75 \ m$ high lighthouse from the sea-level, the angles of depression of two ships are $30^o$ and $45^o$.If one ship is exactly behind the other on the same side of the lighthouse, find the distance between two ships.
→The ratio of the sums of first m and first $n$ terms of an $A.P.$ is $m^2: n^2$. Show that the ratio of its $m ^{\text {th }}$ and $n ^{\text {th }}$ terms is $(2m - 1):(2n -1 ).$
→Show that the following numbers are irrational.
$6+\sqrt{2}$
→$6+\sqrt{2}$
Five cards ten, jack, queen, king and an ace of diamonds are shuffled face downwards. One card is picked at random.
- What is the probability that the card is a queen?
- If a king is drawn first and put aside, what is the probability that the second card picked up is the (i) Ace (ii) King?
Show taht the following points are collinear:A(2, -2), B(-3, 8) and C(-1, 4)
The given distribution shows the number of runs scored by some top batsmen of the world in one-day international cricket matches.
Find the mode of the data.
| Runs scored | Number of batsmen |
| 3000-4000 | 4 |
| 4000-5000 | 18 |
| 5000-6000 | 9 |
| 6000-7000 | 7 |
| 7000-8000 | 6 |
| 8000-9000 | 3 |
| 9000-10000 | 1 |
| 10000-11000 | 1 |
The sides of certain triangles are given below. Determine them are right triangles:
1.6cm, 3.8cm, 4cm.
1.6cm, 3.8cm, 4cm.
Find the sum of all the natural numbers less than $100$ which are divisible by $6.$
→