Exercise 10.1 answers: Circles

Open NCERT p. 147Matches NCERT’s answer

1. A tangent touches the circle at just one point, and that point is called the point of contact.
2. Pick any point on the circle — there is exactly one tangent through it, the line perpendicular to the radius at that point.
3. A circle has infinitely many points, and every one of them gives its own tangent, so the tangents never run out.

AnswerInfinite — a circle can have infinitely many tangents, one through every point of the circle.

Watch this explained “Exactly one tangent, and the normal”, 10:00 into Why the radius meets it at a right angle, argued from shortest distance

Open NCERT p. 147Matches NCERT’s answer

(i) A tangent to a circle intersects it in

1. A tangent, by definition, shares only one point with the circle — that is what makes it different from a line that cuts through.

Answerone

(ii) A line intersecting a circle in two points is called a

1. A line that crosses right through a circle meets it at two points. That kind of line is called a secant.

Answersecant

(iii) A circle can have … parallel tangents at the most

1. Slide a family of parallel lines across a circle. The chord each one cuts shrinks as it moves outward, and it shrinks to zero at exactly one line on each side — so exactly two lines of the family can be tangents.

Answertwo

(iv) The common point of a tangent to a circle and the circle

1. The single point where a tangent touches the circle has its own name.

Answerpoint of contact

Watch this explained “The three names”, 1:51 into Counting shared points sorts every line into one of three kinds

Open NCERT p. 147Checked by computer

1. OP is the radius to the point of contact P, so OP is perpendicular to the tangent PQ.
2. Triangle OPQ is right-angled at P, with OQ as the hypotenuse.
3. By Pythagoras: OQ² = OP² + PQ², so 12² = 5² + PQ².
4. PQ² = 144 − 25 = 119, so PQ = √119 cm.

AnswerPQ = √119 cm (option D).

Watch this explained “Radius five, centre twelve away”, 11:18 into Why the radius meets it at a right angle, argued from shortest distance

Open NCERT p. 147One way to think about it

1. Draw the given line l. Draw a circle with any centre O and any radius r (say 3 cm). Many drawings are correct here: the circle and the line can be anywhere.
2. From O draw the line perpendicular to l, and let it cross the circle at P (either of the two crossing points will do). So OP = r and OP is perpendicular to l.
3. Through P draw a line parallel to l. It is also perpendicular to OP. For any other point Q of this line, triangle OPQ is right-angled at P, so OQ (the hypotenuse) is longer than OP = r, and Q lies outside the circle. The line meets the circle only at P, so it is a tangent parallel to l.
4. Mark any point N on OP strictly between O and P, so ON < r and N is inside the circle. Through N draw a line parallel to l. A line through a point inside a circle crosses the circle at two points, so this line is a secant parallel to l.

In shortOne correct drawing (there are many): the line parallel to l through the end P of the radius perpendicular to l is the tangent, and a line parallel to l through any point between O and P is the secant.

Watch this explained “No turning at all - a family of parallels”, 3:44 into A tangent as what a secant becomes when its two crossings merge