Segment AD is perpendicular to the radius of circle (B) and the radius of circle (c). If radius BA=6in, segment AD=21in and radius CD=16in, find the length of BC to the nearest tenth of an inch.
Find the segment that makes BC the hypotenuse of a right triangle by finding a segment with an endpoint B that is perpendicular to CD. You can do this by subtracting DC and BA to get a segment with the length of 10. This segment is parallel to AD so you know that the length will also be 21. Use Pythagorean Theorem with the values 10 and 21 to find the hypotenuse BC.
10^2 + 21^2 = (BC)^2
100 + 441 = (BC)^2
541 = (BC)^2
square root of 541 = BC
BC is about 23.3 inches
To find the length of BC, we need to apply the Pythagorean theorem to triangle BCD.
In triangle BCD with right angle at D, we have:
BD^2 + CD^2 = BC^2
Since segment AD is perpendicular to both radii BA and CD, it bisects BC. Therefore, segment BD = (BC/2).
Substituting for BD and rearranging the equation, we get:
(BD)^2 + CD^2 = (BC/2)^2
(BC/2)^2 = (BD)^2 + CD^2
We can substitute the known values:
(BD)^2 = (6in)^2 + (21in)^2
(BD)^2 = 36in^2 + 441in^2
(BD)^2 = 477in^2
Simplifying, we have:
BD = √477in
Since BD = BC/2, we can substitute this value into the equation to find BC:
√477in = BC/2
Multiplying both sides of the equation by 2, we get:
2 * √477in = BC
Simplifying the square root, we have:
BC ≈ 2 * 21.8in
BC ≈ 43.6in
Therefore, the length of BC is approximately 43.6 inches rounded to the nearest tenth of an inch.
To find the length of BC, we can use the Pythagorean theorem.
First, let's draw a diagram to visualize the problem.
```
A --- D --- C
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B --- O
```
In the diagram, O represents the center of the circle, and the line segment OC and OD represent the radii of circles B and C, respectively.
Since segment AD is perpendicular to the radii BA and CD, we can form right triangles ABO and CDO.
In right triangle ABO, we have:
AB = 6 inches (given)
AD = 21 inches (given)
AO = AB + BO
To find BO, we can use the Pythagorean theorem:
AB^2 + BO^2 = AO^2
Rearranging the equation, we get:
BO^2 = AO^2 - AB^2
Substituting the given values, we find:
BO^2 = (AB + BO)^2 - AB^2
BO^2 = (6 + BO)^2 - 6^2
Simplifying,
BO^2 = (BO + 6)^2 - 36
BO^2 = BO^2 + 12BO + 36 - 36
BO^2 = BO^2 + 12BO
Subtracting BO^2 from both sides,
0 = 12BO
Since BO cannot be zero, we know that BO = 0 is not a valid solution.
Hence, there is no solution for BO in right triangle ABO.
Now, let's move on to right triangle CDO.
In right triangle CDO, we have:
CD = 16 inches (given)
AD = 21 inches (given)
CO = CD + DO
To find DO, we can use the Pythagorean theorem:
CD^2 + DO^2 = CO^2
Rearranging the equation, we get:
DO^2 = CO^2 - CD^2
Substituting the given values, we find:
DO^2 = (CD + DO)^2 - CD^2
DO^2 = (16 + DO)^2 - 16^2
Simplifying,
DO^2 = (DO + 16)^2 - 256
DO^2 = DO^2 + 32DO + 256 - 256
DO^2 = DO^2 + 32DO
Subtracting DO^2 from both sides,
0 = 32DO
Again, since DO cannot be zero, we know that DO = 0 is not a valid solution.
Therefore, there is no solution for DO in right triangle CDO.
Since we haven't found a valid solution for either triangle, we can conclude that the given information does not provide enough information to determine the length of BC.
Let M be the midpoint of AD.
Then you have two right triangles BMA and DMC where you know the hypotenuse and one leg.
Use the Pythagorean Theorem to find BM and MC.