SOLUTION: the arch of this bridge is an arc of a circle. the arc measures 80 degrees and the bridge spans a 200 meters across a valley. what is the length of the bridge's arch?

Question 1207162: the arch of this bridge is an arc of a circle. the arc measures 80 degrees and the bridge spans a 200 meters across a valley. what is the length of the bridge's arch?
Found 3 solutions by MathLover1, ikleyn, math_tutor2020:
Answer by MathLover1(20850)   (Show Source): You can put this solution on YOUR website!

The length of an arc depends on the of a circle and the central angle .
We know that for the angle equal to degrees (), the arc length is equal to circumference.
Hence, as the proportion between angle and arc length is constant, we can say that:
.....as circumference

given:
=>convert to radians => radians

Answer by ikleyn(52803)   (Show Source): You can put this solution on YOUR website!
.
the arch of this bridge is an arc of a circle. the arc measures 80 degrees
and the bridge spans a 200 meters across a valley.
what is the length of the bridge's arch?
~~~~~~~~~~~~~~~~~~~~~~

Make a sketch. You have an arc with the center O, the central angle of 80 degrees and the endpoints A and B that are endpoint of the bridge. Draw the radius from O, which bisects the arc and intersect the horizontal chord at point C. You have right-angled triangle AOC with one acute angle of 40 degrees and the opposite leg AC of 200/2 = 100 meters long and adjacent leg OC. The radius of the arc is R = = = 155.5936 m (rounded). The length of the bridge is the length of the arc AB and is equal to = = = 217.25 m (rounded). ANSWER. The length of the bridge is about 217.25 meters.

Solved.

===============

The solution in the post by @MathLover1 is INCORRECT.

Answer by math_tutor2020(3817)   (Show Source): You can put this solution on YOUR website!

Draw a circle centered at point A.
Plot points B and C that are on the circle, and that are on the same horizontal level.
Draw segment BC.
Erase a big chunk of the circle so you're only left with minor arc BC.
The arc has endpoints B and C.

Based on the values mentioned in the instructions, we can say that:
Chord BC is 200 meters
Angle BAC = 80 degrees

Let D be the midpoint of segment BC.
This means BD = DC and BD is half of BC
Therefore BD = 200/2 = 100 meters.

Draw a segment AD to form triangles ABD and ACD.
They are congruent right triangles. We can use the SSS (side side side) congruence theorem to prove this claim.

Because we have congruent triangles, we know that
angle DAB = angle DAC = 40
These angles are half of angle BAC = 80.

Let's focus entirely on triangle ABD.
We'll use a trig ratio to determine radius AB.
sin(angle) = opposite/hypotenuse
sin(A) = BD/AB
sin(40) = 100/AB
AB = 100/sin(40)
AB = 155.572382686 approximately is the radius
Please make sure that your calculator is set to degrees mode.

Let's determine the circumference of this circle.
C = 2pi*r
C = 2pi*155.572382686
C = 977.4901090956
I used the calculator's stored version of pi to get the most accuracy possible.

This is the approximate distance around the full circle.
However, we don't want a full circle.
Instead we just want the arc piece.
80/360 = 2/9
We want 2/9 of a full circle.
(2/9)*977.4901090956 = 217.220024243467

Answer: approximately 217.220024 meters
This is the curved distance from B to C.
Round this value however your teacher instructs.
Your answer may vary depending how you rounded intermediate steps and what level of precision you use for pi.

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