Question

In a triangle ABC, angle BAC = 90 degree and AD is perpendicular to BC . Then which one is true and why? :-


BD.CD = BC2

BD.CD = AD2

AB.AC = BC2

AB.AC = AD2

Answer 1

In a triangle ABC, angle BAC = 90 degree and AD is perpendicular to BC then, $\mathrm{BD} \times \mathrm{CD}=\mathrm{AD}^{2}$

Solution:

The figure is attached below

In Triangle ABC, angle BAC = 90 degree and AD is perpendicular to BC

Given in the question a perpendicular is drawn AD on BC from angle BAC which is 90 degree

So, these two triangles  ABD and CAD  are similar

$\Delta \mathrm{ABD} \approx \Delta \mathrm{CAD}$

By Corresponding part of Similar Triangles (CPST),

If two triangles are similar, then the ratio of the area of both triangles is proportional to the square of the ratio of their corresponding sides.

This proves that the ratio of areas of two similar triangles is proportional to the squares of the corresponding sides of both the triangles.

$\text {we can write } \frac{B D}{A D}=\frac{A D}{C D}$

By cross-multiplication we get,

$\mathrm{BD} \times \mathrm{CD}=\mathrm{AD} \times \mathrm{AD}=\mathrm{AD}^{2}$

$\mathrm{BD} \times \mathrm{CD}=\mathrm{AD}^{2}$

Which is our Required Expression

Hence , option 2 is correct

Learn more about triangles

In triangle ABC, angle BAC=90, AD is the bisector of angle BAC and DE is perpendicular to AC. Prove that DEx(AB+AC)=ABxAC.

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In a triangle ABC angle BAC = 90 degree and AD is drawn perp. to BC. Prove that AD2 = BD.CD. Prove it using Pythagoras theorem

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Answer 2

Step-by-step explanation:

(AD}^{2}BD×CD=AD

2

Solution:

In Triangle ABC, angle BAC = 90 degree and AD is perpendicular to BC

Given in the question a perpendicular is drawn AD on BC from angle BAC which is 90 degree

So, these two triangles ABD and CAD are similar

ΔABD≈ΔCAD

By Corresponding part of Similar Triangles (CPST),

If two triangles are similar, then the ratio of the area of both triangles is proportional to the square of the ratio of their corresponding sides.

This proves that the ratio of areas of two similar triangles is proportional to the squares of the corresponding sides of both the triangles.

we can write

AD.BB= CDAD

By cross-multiplication we get,

BD×CD=AD×AD=AD

\mathrm{BD} \times \mathrm{CD}=\mathrm{AD}^{2}BD×CD=AD

2

Which is our Required Expression

Hence , option 2 is correct