Two lengths are drawn from a Point P to a circle centre (o). The radius of the circle is 8.2cm and P is 12.4cm from o. Find the angle between the tangents.
Please help, I really don't understand it.
If one of the lengths goes to the center, where does the other one go? Is is a tangent? You don't say so. You talk about "between the tangents" but one of the lines is not a tangent.
If I understand the question, tangents are drawn to the circle from point P.
Let those contact points be A and B
In triangle APO,
AO=8.2
OP = 12.4
and the angle at A is 90º
let angle APO be ß
sin ß=8.2/12.4
ß = 41.4º
so the angle between the two tangents is 82.8º
To find the angle between the tangents, we can use the properties of tangents to a circle.
First, let's draw the diagram to visualize the problem.
P
|\
| \
_______ | \
| | | \
| o | | \
|_______| __|______\
A B
We have a circle with center "o" and radius 8.2 cm. Point P is located at a distance of 12.4 cm from the center "o". Two tangents are drawn from point P to the circle, intersecting the circle at points A and B.
To find the angle between the tangents, we need to find the angle at point P.
Now, let's use the properties of tangents to find the angle at point P.
1. The angle between the tangent and the radius of a circle at the point of contact is always 90 degrees.
So, angle APO = 90 degrees and angle BPO = 90 degrees.
2. The sum of the angles of a triangle is always 180 degrees.
So, angle AOP + angle BOP + angle APO + angle BPO = 180 degrees.
Since angle APO and angle BPO are both 90 degrees, we can simplify the equation to:
angle AOP + angle BOP + 90 degrees + 90 degrees = 180 degrees.
angle AOP + angle BOP = 180 degrees - 180 degrees.
angle AOP + angle BOP = 0 degrees.
Therefore, angle AOP + angle BOP = 0 degrees.
From the given information, we know that point P is equidistant from points A and B (P is the midpoint of line segment AB).
So, angle AOP = angle BOP. Let's denote this angle as "x".
Therefore, we can rewrite our equation as:
x + x = 0 degrees.
2x = 0 degrees.
Now, solving for x:
x = 0 degrees / 2.
x = 0 degrees.
Thus, the angle between the tangents is 0 degrees.
Therefore, there is no angle between the tangents.