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24 votes
What’s the chord BC?

What’s the chord BC?-example-1
User Welton
by
3.1k points

1 Answer

16 votes
16 votes

Answer:

D) 3.8 cm

Explanation:

There are several ways this problem can be solved. Maybe the easiest is to use the Law of Cosines to find angle BAC. Then trig functions can be used to find the length of the chord.

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In triangle BAC, the Law of Cosines tells us ...

a² = b² +c² -2bc·cos(A)

A = arccos((b² +c² -a²)/(2bc)) = arccos((8² +6² -3²)/(2·8·6)) = arccos(91/96)

A ≈ 18.573°

The measure of half the chord is AB times the sine of this angle:

BD = 2(AB·sin(A)) ≈ 3.82222

The length of the common chord is about 3.8 cm.

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Additional comment

Another solution can be found using Heron's formula to find the area of triangle ABC. From that, its altitude can be found.

Area ABC = √(s(s-a)(s-b)(s-c)) . . . . where s=(a+b+c)/2

s=(3+8+6)/2 = 8.5

A = √(8.5(8.5 -3)(8.5 -8)(8.5 -6)) = √54.4375 ≈ 7.64444

The altitude of triangle ABC to segment AC is given by ...

A = 1/2bh

h = 2A/b = 2(7.64444)/8 = 1.911111

BD = 2h = 3.822222

What’s the chord BC?-example-1
User Kitze
by
3.5k points
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