The vertices of a $\Delta \mathrm{ABC}$ are $\mathrm{A}(4,6), \mathrm{B}(1,5)$ and $\mathrm{C}(7,2)$. A line is drawn to intersect sides
$\mathrm{AB}$ and $\mathrm{AC}$ at $\mathrm{D}$ and $\mathrm{E}$ respectively, such that $\frac{\mathrm{AD}}{\mathrm{AB}}=\frac{\mathrm{AE}}{\mathrm{AC}}=\frac{1}{4}$. Calculate the area of the $\Delta \mathrm{ADE}$ and compare it with the area of $\Delta \mathrm{ABC}$. (Recall Theorem $6.2$ and Theorem 6.6).

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Geometry

NCERT Class 10 - Math

Chapter 7

Coordinate Geometry

Section 4

Area of a Triangle

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Video Transcript

students Welcome back in this question. They are given a triangle triangle abc. And we are given the vortices of the strangle. Ok is given to a four comma six B is given to be 1:05. She is given to be seven comma one. Okay, so now in this question we are given Sorry It's 7:02. Okay, so this question we are given the vortices of this triangle abc. Are this a B, N C? A line is drawn to intercept A B and a C at D and E respectively. Such that a day upon A B is equal to a upon as equals to four. So we are given a condition that Eddie upon A B is equal to E upon a C. Which is equals to one by four. Okay, so we are given this condition now we have to calculate the area of the triangle A. D. E. And then we have to compare it with the area of triangle abc. Okay, so since we're given this condition uh I'll just split this uh equation up. Okay, so let me draw this point B. And E point. Okay, so let's say this is point B. And let's say this is point E. Okay, so if I uh if I rewrite be given equation in our different form, what do I get this 80 divided by B. B is equal to one x 3 night which is equal to E divided by easy. Right because what we are given is that 80? This part is 1/4 of the baby. Okay, since this this small part is 1/4 of this whole part then. What we can say that this part Is 1/3 of this part? Right so I have written D divided by D. B. Is equal to one by three. Now we have to find out the vortices of the. Okay the coordinates of the so for the X will be equal to mx two plus an X. One divided by m plus end. Okay I have applied section formula. So if I put the values what do I get? This one multiplied by one plus three, multiplied by four divided by three plus one. Okay so uh the X coordinate of D comes out to be entertained by four. Similarly finding out the y coordinate what do I get us? M. Y two plus and violent divided by M plus. And so if you put the values what will you get us? 23 by four. Okay so I can say that the coordinates of D has come out to be equal to 13 x four and 23 x four. Similarly, I will find out the coordinates of E. Okay so finding out the coordinates of E. So for E what do I get this coordinate of X will be one times of seven plus three times of four divided by four. That will be 19 by four. Okay and similarly why? According it will be one multiplied by two plus three multiplied by six and divide it by four. Okay that means five. So what we can say the coordinates of E Coinates of e. s. 19 x four Coma five. Okay. So now we can find out the area of A. B. C. As well as the area of A. D. E. So area of triangle abc would be equal to half of four times of 5 -2 Plus one times off two minus six plus seven times of six minus. Right? Right so the answer would come out of the 15 by two and similarly if I find out the area of area of triangle Eddie yeah then it would come out to be half of four times off 23 x 4 -5 Plus 13 x four. Five minus six plus 19 by four. That's crazy. 6 -23 x four. Right so this would come out to be quanto nine by four. So since we have to compare these areas so what I will do is I'll find the ratio of these two areas. Okay so the ratio of area of triangle A. D. E. Divided by the area off triangle A. B. C. would be equal to nine x 4 divided by 15 by That would come out to be three x 10. So this is our answer. Thank you

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