X, Y and Z are points on the sides AB, BC and AC of the triangle ABC, such that AX:XB =4:3, BY:YC=2:3, CZ:ZA=2:1. Find the ratio of the area of the triangle XYZ to that of triangle ABC.


Question and Answers Forum

All Questions Topic List
Geometry Questions
Previous in All Question Next in All Question
Previous in Geometry Next in Geometry
Question Number 208158 by necx122 last updated on 06/Jun/24

$X, Y a n d Z a r e p o i n t s o n t h e s i d e s A B,$ $B C a n d A C o f t h e t r i a n g l e A B C, s u c h$ $t h a t A X : X B = 4 : 3, B Y : Y C = 2 : 3,$ $C Z : Z A = 2 : 1 . F i n d t h e r a t i o o f t h e a r e a$ $o f t h e t r i a n g l e X Y Z t o t h a t o f t r i a n g l e$ $A B C .$
	Answered by mr W last updated on 06/Jun/24

	Commented by mr W last updated on 06/Jun/24

	$\frac{Δ A X Z}{Δ A B C} = \frac{4 \times 1}{(4 + 3) \times (1 + 2)} = \frac{4}{21}$ $\frac{Δ B Y X}{Δ A B C} = \frac{3 \times 2}{(3 + 4) \times (2 + 3)} = \frac{6}{35}$ $\frac{Δ C Z Y}{Δ A B C} = \frac{3 \times 2}{(3 + 2) \times (2 + 1)} = \frac{6}{15}$ $\frac{Δ X Y Z}{Δ A B C} = \frac{Δ A B C - Δ A X Z - Δ B Y X - Δ C Z Y}{Δ A B C}$ $= 1 - \frac{4}{21} - \frac{6}{35} - \frac{6}{15} = \frac{5}{21}$
	Commented by Ari last updated on 07/Jun/24

	Commented by mr W last updated on 07/Jun/24

	Commented by mr W last updated on 07/Jun/24

	$Δ A D E = \frac{a \times b \times \sin θ}{2}$ $Δ A B C = \frac{(a + c) \times (b + d) \times \sin θ}{2}$ $\Rightarrow \frac{Δ A D E}{Δ A B C} = \frac{a \times b}{(a + c) \times (b + d)}$
	Commented by necx122 last updated on 07/Jun/24

	$T h i s i s r e a l l y c o o l .$
	Commented by Ari last updated on 08/Jun/24

	$t h a n k y o u S i r!$
	Commented by Tawa11 last updated on 21/Jun/24

	$Weldone sir$
Terms of Service
Privacy Policy
Contact: info@tinkutara.com

Question and Answers Forum