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In Fig. 6.18, ABC is a triangle right angled at B and BD ⊥ AC. If AD = 4 cm, and CD = 5 cm, find BD and AB

Solution:

Given, ABC is a right triangle with B at right angle.

Also, BD ⊥ AC

The length of the sides

AD = 4 cm

CD = 5 cm

We have to find the length of the sides BD and AB.

In △ADB and △CDB,

∠BDA = ∠BDC = 90°

∠BAD = ∠CBD = 90° - ∠C

AAA criterion states that if two angles of a triangle are respectively equal to two angles of another triangle, then by the angle sum property of a triangle their third angle will also be equal.

By AAA, △ADB ⩬ △CDB

By the property of similarity,

The corresponding sides are proportional.

So, BD/AD = CD/BD = AB/BC

Considering BD/AD = CD/BD

BD² = AD . CD

BD² = 4 . 5

BD² = 20

Taking square root,

BD = 2√5 cm

In △BDC,

By pythagoras theorem,

CB² = BD²+ CD²

CB² = (2√5)²+ (5)²

CB²= 20 + 25

CB² = 45

Taking square root,

CB = 3√5 cm

Considering BD/CD = AB/BC,

2√5/5 = AB/3√5

5AB = 2√5(3√5)

5AB = 6(5)

AB = 30/5

AB = 6 cm

Therefore, the length of sides AB and BD is 6 cm and 2√5 cm.

✦ Try This: ABC is a triangle right angled at B and BD ⊥ AC. If AD = 8 cm, and CD = 12 cm, find BD and AB.

☛ Also Check: NCERT Solutions for Class 10 Maths Chapter 6

NCERT Exemplar Class 10 Maths Exercise 6.4 Problem 9

In Fig. 6.18, ABC is a triangle right angled at B and BD ⊥ AC. If AD = 4 cm, and CD = 5 cm, find BD and AB

Summary:

In Fig. 6.18, ABC is a triangle right angled at B and BD ⊥ AC. If AD = 4 cm, and CD = 5 cm, then BD = 2√5 cm and AB = 6 cm

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