The tangent from point T touches a circle at R.If the radius of the circle i 3cm and T is 5cm from the center of the circle.Calculate/TR/.
To calculate the length of TR, we can use the properties of tangents and radii in a circle.
We know that the radius of the circle is 3 cm, and T is 5 cm from the center of the circle. Using Pythagoras' theorem, we can find the length of TR.
First, draw a diagram to better visualize the problem. Label the center of the circle as O, the point where the tangent touches the circle as R, and the point where the tangent is extended to meet the radius as T.
Since T is 5 cm from the center of the circle and TR is perpendicular to the radius, we can form a right-angled triangle OTT'.
The hypotenuse OT represents the radius of the circle, which is 3 cm. The perpendicular side is TR, and the base of the triangle is TT'.
Using Pythagoras' theorem, we have:
OT^2 = TR^2 + TT'^2
Since OT is the radius and it is 3 cm, we can substitute it into the equation:
3^2 = TR^2 + TT'^2
Simplifying the equation:
9 = TR^2 + TT'^2
Since TR is the length we want to find, we can rearrange the equation to isolate TR:
TR^2 = 9 - TT'^2
To find TT', we can use the fact that the tangent from point T is perpendicular to the radius. This means that OTT' is a right-angled triangle, and we can use the Pythagorean theorem again to find TT'.
Since T is 5 cm from the center of the circle and the radius is 3 cm, we have:
TT'^2 = OT^2 - OT'^2
TT'^2 = 5^2 - 3^2
TT'^2 = 25 - 9
TT'^2 = 16
Substituting this value back into the equation for TR:
TR^2 = 9 - TT'^2
TR^2 = 9 - 16
TR^2 = -7
Since the square of a length cannot be negative, it means that TR is an imaginary or complex number. Therefore, we cannot calculate the length of TR using the given information.
Draw a diagram for a geometry problem, and half of the time the solution will be right in front of your eyes.
Let the circle C with radius r has a centre O and the external point R is distance d away from O.
Draw a tangent from R to circle C and tangent at T.
Join OT.
By definition of a tangent, angle OTR is a right angle.
OT is known (r)
OR is known (d)
We need to calculate the distance TR, which is the remaining side of the right triangle OTR.
Bet you can take it from here.