In a triangle ABD, C is the midpoint of BD. If AB=10, AD=12, AC=9, find the value of BD?
Answers
BD = 2√41 = 12.8 cm if In a Δ ABD, C is the midpoint of BD & AB=10, AD=12, AC=9
Step-by-step explanation:
Let say side BD = 2a
Then find Area of Δ ABD
s = (10 + 12 + 2a)/2 = 11 + a
Area = √(11 + a)(a + 1)(a - 1)(11 - a)
= √( 121 - a²)(a² - 1)
as AC is median
Area of ΔABC = Area of ΔACD = (1/2) Area of Δ ABD
find Area of ΔABC
s = (10 + 9 + a)/2 = (19 + a)/2
Area of ΔABC = √((19 + a)(a - 1)(a + 1)(19 - a) /(2 * 2 * 2 * 2))
= √(361 - a²)(a² - 1)/16
√(361 - a²)(a² - 1)/16 = (1/2) √( 121 - a²)(a² - 1)
Squaring both sides
=> (361 - a²)(a² - 1)/16 = (1/4) ( 121 - a²)(a² - 1)
=> 361 - a² = 4( 121 - a²)
=> 3a² = 123
=> a² = 41
=> a = √41
=> 2a = 2√41
BD = 2√41 = 12.8 cm
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