Constructions Test - 4

With the help of a ruler and a compass, we can construct an angle which is a multiple of 15⁰. Since 7.5⁰ is multiple of 15⁰, so, we can construct it.

With the help of a ruler and a compass, it is not possible to construct an angle which is not a multipe of 15⁰. Since 40⁰ ia not a multiple of 15⁰, so, we cannot construct it.

Two circles are congruent if they are of same size. The size can be measured as the radii, diameter. So, if two circles have same radii or diameter, then those circles are congruent.

Concentric circles are those which have same centres but different radii or diameters.

Consider, two circles with Centre A & B touches internally at point P,

Now distance between their Centre i.e. AB can be given as,    AB= AP - BP, where,

AP= radius of circle A ;  BP= radius of circle B 

Thus, we can say that the distance between the centers of two circles touching internally is equal to the difference of their radii.

Consider a circle with Centre O and radius 'r', AB is the tangent on circle O touching the circle at point P. Join OP such that OP becomes the radius of circle O. Now let us take a point Q , other than P, on the tangent AB. Join OQ.  Since Q is a point on tangent AB, other than the point of contact P, so Q will be outside the circle.

Let OQ intersect the circle at R.

Then OQ = OR + RQ

⟹OQ > OR

⟹OQ > OP ( OR=OP=RADIUS)

Thus, OP is the shortest distance among all line segments which is thus perpendicular from O on AB.

Hence, OP is perpendicular to AB.

The radius of any circle is defined as the distance of any point on the circumference of the circle from the Centre of the circle. Since, Centre of any circle is only one. 

Thus, any radius will always be drawn from Centre , hence, both radii will meet and inclined at an angle.

At first we will have to draw the diagonal and then by taking end points of diagonals as the Centre we can draw the arcs corresponding to the length of the four sides given. Therefore, the measures of four sides and one diagonal is necessary for the construction of a quadrilateral.