Question

5. A line parallel to the side BC of triangle ABC intersects the sides AB and AC respectively at the points X and Y respectively. If AX = 2.4 cm , AY = 3.2 cm and YC =4.8 cm then length of AB=​

Answer 1

$Given, \\ \\ AY\:=\:3.2\:cm \\ \\ YC \:=\:4.8 \:cm \\ \\ AX\:=\:2.4\:cm \\ \\ To \:find, \\ \\ XB \:and\: AB \\ \\ \\ Solution , \\ \\ \\ By\:Basic\: Proportionality\:Theorem \\ \\ \frac{AX}{XB}\:=\: \frac{AY}{YC} \\ \\ Here \:XB\:=\: \frac{4.8}{3.2} \times 2.4 \\\\ XB\:=\:3.6\:cm. \\ \\ Now \:=\:XB\:+\:AX\:=\:2.4+2.6\:=\:6 \\ AX\:=\:6\:cm \\ \\ \color{aqua}{ \underline{XB\:=\:3.6\:cm\:and\:AB\:=\:6\:cm}}$

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Basic Proportionality Theorem :-

This theorem states that if a line is drawn parallel to one side of a triangle, intersects the other two lines in distinct points, then the other two sides are divided in the same ratio.

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MORE Theorems to know :-

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1. Pythagoras Theorem :-

This theorem states that : In a right-angled triangle, the square of the longest side is equal to sum of the squares of remaining sides.

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2. Converse of Pythagoras Theorem :-

This theorem states that : If the square of the longest side is equal to sum of the squares of remaining two sides, angle opposite to longest side is right angle.

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3. Area Ratio Theorem :-

This theorem states that :- The ratio of the area of two similar triangles is equal to the ratio of the squares of corresponding sides.

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Answer 2

$\large\underline{\sf{Given- }}$

A line parallel to the side BC of triangle ABC intersects the sides AB and AC respectively at the points X and Y respectively.

AX = 2.4 cm , AY = 3.2 cm and YC = 4.8 cm

$\large\underline{\sf{To\:Find - }}$

Length of AB

$\large\underline{\sf{Solution-}}$

Given that,

A line parallel to the side BC of triangle ABC intersects the sides AB and AC respectively at the points X and Y respectively and AX = 2.4 cm , AY = 3.2 cm and YC = 4.8 cm.

We know,

Basic Proportionality Theorem :- This theorem states that if a line is drawn parallel to one side of a triangle, intersects the other two lines in distinct points, then the other two sides are divided in the same ratio.

So, By using Basic Proportionality Theorem, we have

$\rm \: \dfrac{AX}{XB} = \dfrac{AY}{YC}$

can also be rewritten as

$\rm \: \dfrac{XB}{AX} = \dfrac{YC}{AY}$

On adding 1 in each term, we get

$\rm \: \dfrac{XB}{AX} + 1 = \dfrac{YC}{AY} + 1$

$\rm \: \dfrac{XB + AX}{AX} = \dfrac{YC + AY}{AY}$

$\rm \: \dfrac{AB}{AX} = \dfrac{AC}{AY}$

On substituting the values, we get

$\rm \: \dfrac{AB}{2.4} = \dfrac{8}{3.2}$

$\rm \: \dfrac{AB}{24} = \dfrac{8}{32}$

$\rm \: \dfrac{AB}{24} = \dfrac{1}{4}$

$\rm \: {AB}= \dfrac{24}{4}$

$\rm\implies \:\rm \: {AB}= 6 \: cm$

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MORE TO KNOW

1. Pythagoras Theorem :-

This theorem states that : In a right-angled triangle, the square of the longest side is equal to sum of the squares of remaining sides.

2. Converse of Pythagoras Theorem :-

This theorem states that : If the square of the longest side is equal to sum of the squares of remaining two sides, angle opposite to longest side is right angle.

3. Area Ratio Theorem :-

This theorem states that :- The ratio of the area of two similar triangles is equal to the ratio of the squares of corresponding sides.