Question

Draw a pair of tangents of radius $2cm$ that are inclined to each other at an angle of $90$ degrees.

Answer 1

1) Draw a circle of radius $2cm$ and draw a horizontal radius PO where O is the center and P is a point on the circle.

(2) Draw ${90}^{\circ }$ from point O such that the ray of an angle intersects the circle at R.

(3) Now at P, draw ${90}^{\circ }$

(4) Now, at R, draw ${90}^{\circ }$

(5) Where the two arcs intersect, mark it as point Q. And thus PQ and PR are two tangents at an angle ${90}^{\circ }$

Now to prove angle between PQ and QR is ${90}^{\circ }$.

then in quadrilateral OPQR, sum of angles $={360}^{\circ }$

$\therefore \angle P+\angle Q+\angle R+\angle O={360}^{\circ }$

${90}^{\circ }+{90}^{\circ }+{90}^{\circ }+\angle Q={360}^{\circ }$

${270}^{\circ }+\angle Q={360}^{\circ }$

$\angle Q={360}^{\circ }-{270}^{\circ }={90}^{\circ }$

$\angle Q={90}^{\circ }$

Hence the angle between PQ and PR is ${90}^{\circ }$

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