Given: Triangle ABC, AD- bisector, AB = 4cm, AC = 8cm, BC = 6cm, Find: BD, CD. Sabc: Sabd.

| April 17, 2021 | education

Let a triangle ABC be given, AD be a bisector,
Since, by the property of the bisector, it divides the opposite side into parts proportional to the adjacent sides, then BD: CD = AB: AC. We denote the length of the segment BD = x cm, since, by condition, AB = 4 cm, AC = 8 cm, BC = 6 cm, then CD = 6 – x, we get the proportion: x: (6 – x) = 4: 8 ; x = 2 (cm); 6 – x = 6 – 2 = 4 (cm) or BD = 2 cm; CD = 4 cm. To find the area ratio S (ABC): S (ABD), we use Heron’s formula.
Let us denote the lengths of the sides of the triangle ABC by the letters a, b and c, the semiperimeter p = (a + b + c): 2, p = (4 + 8 + 6): 2 = 9 (cm), then S (ABC) ^ 2 = p (p – a) (p – b) (p – c), S (ABC) ^ 2 = 9 ∙ 5 ∙ 1 ∙ 3 = 135.
Let’s find the length of the bisector: AD ^ 2 = AB ∙ AC – BD ∙ CD = 4 ∙ 8 – 2 ∙ 4 = 24; AD = (24) ^ (1/2).
For a triangle ABD, we denote the lengths of the sides by the letters a, m and d, then p2 = (a + m + d): 2, p2 = (4 + (24) ^ (1/2) + 2): 2, S (ABD) ^ 2 = p2 (p2 – a) (p2 – b) (p2 – d), S (ABD) ^ 2 = 15. We get S (ABC): S (ABD) = 3. Answer: BD = 2 cm; CD = 4 cm; S (ABC): S (ABD) = 3.



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