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Right Triangle Example

Question: A team is going to build a bridge across a river. Before building the bridge, the team need to measure the width of the river. They make a pole marked as point A. In other side of the river, they make the second pole marked as point B. In the same side of the river with point B, they make a pole marked as point C. Then they measured the angle ABC which is 45 degrees, the distance from B to C is 1000 feet, and the angle ACB is 30 degrees. Then the team can measure the width of the river. What is the width of the river?

Solution:

Let AD be x,

because AD is perpendicular to BC, so triangle ABD and triangle ACD are right triangles.

In right triangle ABD, because the angle B is 45o, so angle BAD = 90o - 45o = 45o

so, AD = BD = x (In a triangle, congruent angles opposite congruent sides.)

Because BD + DC = BC = 1000, so DC = 1000 - BD = 1000 - x

In right triangle ADC, angle DAC = 90o - 30o = 60o

tan DAC = DC/AD, so tan 60o = (1000 - x)/x

Because tan 60o = (square root of 3),
so, (1000 - x)/x = (square root of 3)
(square root of 3) x = 1000 - x
(square root of 3) x + x = 1000
x (square root of 3 + 1) = 1000
x = 1000/(square root of 3 + 1)
= 1000 (square root of 3 - 1)/(square root of 3 + 1) (square root of 3 - 1)
= 1000 (square root of 3 - 1)/(3 - 1)
= 1000 (1.732 - 1)/2
= 1000 (0.732)/2 = 732/2 = 366 feet
so, the width of the river is 366 feet.