Question

D and E are respectively the points on the sides AB and ACof a triangle ABC such that AE = 5 cm AC. = 7.5 cmDE = 4.2 cm and DE || BC length of BC is equal to ​

Answer 1

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

Given that,

D and E are respectively the points on the sides AB and AC of a triangle ABC such that

• AE = 5 cm

• AC. = 7.5 cm

• DE = 4.2 cm

• DE || BC

Now,

$\purple{\rm :\longmapsto\:In \: \triangle \: ADE \: and \: \triangle \: ABC}$

$\purple{\rm :\longmapsto\:\: \angle \: ADE \: and \: \angle \: ABC \: \: \: \{corresponding \: angles \}}$

$\purple{\rm :\longmapsto\:\: \angle \: AED \: and \: \angle \: ACB \: \: \: \{corresponding \: angles \}}$

$\purple{\rm\implies \: \triangle \: ADE \sim \: \triangle \: ABC \: \: \: \{AA \: similarity\}}$

$\purple{\rm\implies \:\dfrac{AE}{AC} = \dfrac{DE}{BC} }$

$\purple{\rm\implies \:\dfrac{5}{7.5} = \dfrac{4.2}{BC} }$

$\purple{\rm\implies \:\dfrac{1}{1.5} = \dfrac{4.2}{BC} }$

$\purple{\rm\implies \:BC = 4.2 \times 1.5 = 6.3 \: cm }$

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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.

4. 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.

Answer 2

Question:-

D and E are respectively the points on the sides AB and ACof a triangle ABC such that AE = 5cm,

AC = 7.5cm DE = 4.2cm and DE || BC length of BC is equal to.

Given:-

D is a point on AB, E is the point on AC of △ABC.

• AE = 5cm.

• AC = 7.5cm.

• DE = 4.2cm.

• DE || BC.

To Find:-

• Length of BC.

solution:-

DE || BC.

⟼ ∠ADE = ∠ABC (Corresponding angles).

⟼ ∠AED = ∠ACB (Corresponding angles).

Hence, by AA criterion of similarly.

△ADE ~ △ABC

Ratio of Corresponding sides of similar triangles are equal.

$\sf \red{⟹ \frac{AE}{AC} = \frac{DE}{BC} }$

$\sf \red{⟹ \frac{5}{7.5} = \frac{4.2}{BC} }$

$\sf \red{⟹ \frac{1}{1.5} = \frac{4.2}{BC} }$

$\sf \red{⟹BC = 4.2 × 1.5 = 6 3cm.}$

Answer:-

$\sf \red{ \therefore \: Length \: of \: BC = 6.3cm.}$

Hope you have satisfied. ⚘