The radius of the base of a cylindrical silo is 28 feet. If the distance from point B to the silo is 53 feet, what is the distance from point B to point A, a point of tangency?

Having seen this problem many many times, consider a top view with O the centre of the silo

then OB = 28+53 = 81 ft
and OA = 28 ft
we have a right-angled triangle, since a tangent meets its radius at 90°
BA^2 + 28^2 = 81^2
BA^2 = 5777
BA=76.007 or appr 76 ft

To find the distance from point B to point A, a point of tangency, we need to use the concept of right triangles and the Pythagorean theorem.

Let's draw a diagram to better understand the situation. The base of the cylindrical silo forms a circle with a radius of 28 feet. Point A represents the point of tangency, and point B is a point outside the circle.

We can draw a line segment from the center of the circle (O) to point A, which represents the radius of the circle. Since point A is a point of tangency, this line segment is perpendicular to line segment AB. This forms a right triangle OAB.

The distance from point B to the center of the circle (O) is the radius of the circle, which is 28 feet.

Since point B is outside the circle, we can draw a perpendicular line from point B to line segment OA. Let's call the point where this line intersects line segment OA as point C.

Now we have another right triangle OBC, where line segment OC is perpendicular to line segment BC.

We know the distance from point B to the silo (the hypotenuse of triangle OBC) is 53 feet.

Now, we can use the Pythagorean theorem to find the distance from point B to point A.

The Pythagorean theorem states that in a right triangle, the square of the length of the hypotenuse (c) is equal to the sum of the squares of the other two sides (a and b). In this case, we are trying to find the length of side a, which is the distance from point B to point A.

Using the Pythagorean theorem, we have:

a^2 + 28^2 = 53^2

Simplifying the equation:

a^2 + 784 = 2809

Subtracting 784 from both sides:

a^2 = 2809 - 784

a^2 = 2025

Taking the square root of both sides:

a = √2025

a ≈ 45

Therefore, the distance from point B to point A, a point of tangency, is approximately 45 feet.

To find the distance from point B to point A, a point of tangency, we need to use the concept of tangents and right triangles.

First, let's draw a diagram to visualize the problem. Draw a circle to represent the base of the cylindrical silo, label the center of the circle C, and mark the points B and A as shown below:

A
|\
| \
| \
| \
|O__B \
C
(center of circle)

The segment OB represents the distance from point B to the center of the circle (silo). The segment OA represents the distance from point A to the center of the circle.

We can see that triangle OCB is a right triangle, with CB as the hypotenuse, and OC and OB as the legs. Since OC is the radius of the base of the silo and given as 28 feet, we know that OC = 28 feet.

Now, we apply the Pythagorean theorem to find the length of OB.

Pythagorean theorem:
(OC)^2 + (OB)^2 = (CB)^2

Substituting the known values:
(28)^2 + (OB)^2 = (53)^2

Simplifying the equation:
784 + (OB)^2 = 2809

To solve for (OB)^2, subtract 784 from both sides:
(OB)^2 = 2809 - 784
(OB)^2 = 2025

To find OB, take the square root of both sides:
OB = sqrt(2025)
OB ≈ 45

Therefore, the distance from point B to point A, a point of tangency, is approximately 45 feet.