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AD is the angle bisector of angle 𝑪𝑨𝑩, so by the angle bisector theorem, 𝑪𝑫: 𝑩𝑫 = 𝑪𝑨: 𝑩𝑨. 3/4 = CA/6 4×CA = 18 CA = 4.5 cm AF is the angle bisector of angle ...

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Byju's Answer

Standard X

Mathematics

Internal Angle Bisector Theorem

If 𝑪𝑫 = 𝟑c...

Question

If 𝑪𝑫 = 𝟑cm, 𝑫𝑩 = 𝟒 cm, 𝑩𝑭 = 𝟒cm, 𝑭𝑬 = 𝟓 cm, 𝑨𝑩 = 𝟔 cm and ∠𝑪𝑨𝑫 ≅ ∠𝑫𝑨𝑩 ≅ ∠𝑩𝑨𝑭 ≅ ∠𝑭𝑨𝑬, find the perimeter of quadrilateral 𝑨𝑬𝑩𝑪 .

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Solution

AD is the angle bisector of angle 𝑪𝑨𝑩, so by the angle bisector theorem, 𝑪𝑫: 𝑩𝑫 = 𝑪𝑨: 𝑩𝑨.

$\frac{3}{4} = \frac{C A}{6}$

4 $\times$ CA = 18

CA = 4.5 cm

AF is the angle bisector of angle 𝑩𝑨𝑬, so by the angle bisector theorem, 𝑩𝑭:𝑬𝑭 = 𝑩𝑨:𝑬𝑨.

$\frac{4}{5} = \frac{6}{E A}$

4 $\times$ EA = 30

EA = 7.5 cm

The perimeter of quadrilateral
𝑨𝑬𝑩𝑪 = 𝑪𝑫 + 𝑫𝑩 + 𝑩𝑭 + 𝑭𝑬 + 𝑬𝑨 + 𝑨𝑪
𝑨𝑬𝑩𝑪 = 𝟑 + 𝟒 + 𝟒 + 𝟓 + 𝟕. 𝟓 + 𝟒. 𝟓
𝑨𝑬𝑩𝑪 = 28 cm

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