The length of the congruent sides of an isosceles triangle is 17cm. If the base of
the triangle is 16 cm, find the length of the median.
18 cm
8 cm
15 cm
8.5 cm
Solve by apollonius theroem
Answers
Answer: Given: ∆ABC is an isosceles triangle. G is the centroid. AB = AC = 13 cm, BC = 10 cm. To find: AG Construction: Extend AG to intersect side BC at D, B – D – C. Centroid G of ∆ABC lies on AD ∴ seg AD is the median. (i) ∴ D is the midpoint of side BC. ∴ DC = 1/2 BC = 1/2 × 10 = 5 In ∆ABC, seg AD is the median. [From (i)] ∴ AB2 + AC2 = 2AD2 + 2DC2 [Apollonius theorem] ∴ 132 + 132 = 2AD2 + 2(5)2 ∴ 2 × 132 = 2AD2 + 2 × 25 ∴ 169 = AD2 + 25 [Dividing both sides by 2] ∴ AD2 = 169 – 25 ∴ AD2 = 144 ∴ AD = √144 [Taking square root of both sides] = 12 cm We know that, the centroid divides the median in the ratio 2 : 1. ∴ AG/GD = 2/1 ∴ GD/AG = 1/2 [By invertendo] ∴ (GD + AG)AG = (1 + 2)/2 [By componendo] ∴ AD/AG = 3/2 = [A – G – D] ∴ 12/AG = 3/2 ∴ AG = (12 x 2)/3 = 8cm ∴ The distance between the vertex oppesite to the base and the centroid id 8 cm