To construct the tangents from a point to a circle (where O is the centre of the circle and P is the point from which tangents are drawn.), following are the steps of construction in the first list named as “Steps” and the reasons behind those construction in the next list named “Reasons”. Match them accordingly.
STEPS:
i) Join OP.
ii) Bisect the segment OP, at L.
iii) Draw a circle with centre as L and radius LO.
iv) From P, join wherever this circle intersects the original circle.
REASONS:
a) Because angle in a semicircle is 90∘ and radius is perpendicular to tangent.
b) We get the point(s) of contact, if they exist.
c) To check where the point P lies and hence determining if tangents exist or not.
d) We need to draw a circle with diameter as the distance between the centre of the circle and the point from where tangents should be drawn.
To construct the tangents from a point to a circle (where O is the centre of the circle and P is the point from which tangents are drawn.), following are the steps of construction in the first list named as “Steps” and the reasons behind those construction in the next list named “Reasons”. Match them accordingly.
STEPS:
i) Join OP.
ii) Bisect the segment OP, at L.
iii) Draw a circle with centre as L and radius LO.
iv) From P, join wherever this circle intersects the original circle.
REASONS:
a) Because angle in a semicircle is 90∘ and radius is perpendicular to tangent.
b) We get the point(s) of contact, if they exist.
c) To check where the point P lies and hence determining if tangents exist or not.
d) We need to draw a circle with diameter as the distance between the centre of the circle and the point from where tangents should be drawn.
The correct option is D i-c, ii-d, iii-a, iv-b
We present here the steps of construction and the reasons behind them.
Step I : Join OP.
Reason: By joining OP, we are determining the location of P. If OP > radius, 2 tangents can be drawn. If OP = radius, just 1 tangent can be drawn and if OP < radius of the circle, i.e. if OP doesn’t cut the circle anywhere, no tangent can be drawn.
Step II : Bisect the segment OP at L.
Reason : To draw a tangent, we need to consider the fact that the radius is perpendicular to the tangent at the point of contact. We also recall that angle in a semicircle is 90°. The idea is to draw a circle such that the diameter is the distance OP and radius of the original circle is
perpendicular to the line joining the end point of the radius and the point P (because it forms the angle in a semi-circle). Refer to the figure. This becomes the tangent.
So the reason to bisect OP at L is so that we can draw a circle with centre as L and radius LO.
Step III : Draw a circle with centre L and radius LO = LP.
Reason : The reason has been explained above as to why we need a circle.
Step IV : From P, join the points wherever this circle intersects the original circle.
Reason : The number of points of intersection is actually the number of contact points. IF OP > radius, we will have two points of intersection, if OP = radius, we will have touching circles, and if OP < radius, we will have non-touching circles and no tangent can be drawn.
We present here the steps of construction and the reasons behind them.
Step I : Join OP.
Reason: By joining OP, we are determining the location of P. If OP > radius, 2 tangents can be drawn. If OP = radius, just 1 tangent can be drawn and if OP < radius of the circle, i.e. if OP doesn’t cut the circle anywhere, no tangent can be drawn.
Step II : Bisect the segment OP at L.
Reason : To draw a tangent, we need to consider the fact that the radius is perpendicular to the tangent at the point of contact. We also recall that angle in a semicircle is 90°. The idea is to draw a circle such that the diameter is the distance OP and radius of the original circle is
perpendicular to the line joining the end point of the radius and the point P (because it forms the angle in a semi-circle). Refer to the figure. This becomes the tangent.
So the reason to bisect OP at L is so that we can draw a circle with centre as L and radius LO.
Step III : Draw a circle with centre L and radius LO = LP.
Reason : The reason has been explained above as to why we need a circle.
Step IV : From P, join the points wherever this circle intersects the original circle.
Reason : The number of points of intersection is actually the number of contact points. IF OP > radius, we will have two points of intersection, if OP = radius, we will have touching circles, and if OP < radius, we will have non-touching circles and no tangent can be drawn.
