SATHEE: knowledge-route Maths10 Cha2

knowledge-route Maths10 Cha2

title: “Lata knowledge-route-Class10-Math1-2 Merged.Pdf(1)” type: “reveal” weight: 1

CONSTRUCTION

10.1 DIVISION OF A LINE SEGENT :

In order to divide a line segment internally is a given ratio $m$ : $n$ , where both $m$ and $n$ are positive integers, we follow the following steps:

CONSTRUCTION

Step of construction :

(i) Draw a line segment $A B$ of given length by using a ruler.

(ii) Draw and ray $A X$ making an acute angle with $A B$ .

(iii) Along $A X$ mark off $(m + n)$ points $A_{1}, A_{2}, \dots, A_{m + n}$ such that $A A_{1} = A_{1} A_{2} = \dots = A_{m + n + n} A_{m + n}$ .

(iv) Join $B A_{m + n}$

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(v) Through the point $A_{m}$ draw a line parallel to $A_{m + n} B$ by making an angle equal to $∠ A A_{m + n} B$ at $A_{m}$ . Suppose this line meets $A B$ at a point $P$ .

The point $P$ so obtained is the required point which divides $A B$ internally in the ratio $m : n$ .

CONSTRUCTION

Ex. 1 Divide a line segment of length $12 c m$ internally in the ratio $3 : 2$ .

CONSTRUCTION

Sol. Following are the steps of construction.

Step of construction :

(i) Draw a line segment $A B = 12 c m$ by using a ruler.

(ii) Draw any ray making an acute angle $∠ B A X$ with $A B$ .

(iii) Along $A X$ , mark-off $5 (= 3 + 2)$ points $A_{1}, A_{2}, A_{3}, A_{4}$ and $A_{5}$ such that $A A_{1} = A_{1} A_{2} = A_{2} A_{3} = A_{3} A_{4} =$ $A_{4} A_{5}$ .

CONSTRUCTION

(iv) Join $B A_{5}$

(v) Through $A_{3}$ draw a line $A_{3} P$ parallel to $A_{5} B$ by making an angle equal to $∠ A A_{5} B$ at $A_{3}$ intersecting $A B$ at a point $P$ .

The point $P$ so obtained is the required point, which divides $A B$ internally in the ratio $3 : 2$ .

CONSTRUCTION

10.2 ALTERNATIVE METHOD FOR DIVISION OF A LINE SEGMENT INTERNALLY IN A GIVEN RATIO :

Use the following steps to divide a given line segment $A B$ internally in a given ration $m : n$ , where $m$ and natural members.

CONSTRUCTION

Steps of Construction :

(i) Draw a line segment $A B$ of given length.

(ii) Draw any ray $A Z$ making an acute angle $∠ B A X$ with $A B$ .

(iii) Draw a ray $B Y$ , on opposite side of $A X$ , parallel to $A X$ making an angle $∠ A B Y$ equal to $∠ B A X$ .

(iv) Mark off a points $A_{1}, A_{2}, \dots . A_{m}$ on $A X$ and $n$ points $B_{1}, B_{2}, \dots B_{n}$ on $B Y$ such that $A A_{1} = A_{1} A_{2} =$ $A_{m - 1} A_{m} = B_{1} B_{2} = \dots . B_{n - 1} B_{n}$ .

(v) Join $A_{m} B_{n}$ . Suppose it intersect $A B$ at $P$ .

CONSTRUCTION

The point $P$ is the required point dividing $A B$ in the ratio $m : n$ .

CONSTRUCTION

Ex. 2 Decide a line segment of length $6 c m$ internally in the ratio $3 : 4$ .

CONSTRUCTION

Sol. Follow the following steps :

Steps of Construction :

(i) Draw a line segment $A B$ of length $6 c m$ .

(ii) Draw any ray $A X$ making an acute angle $∠ B A X$ with $A B$ .

(iii) Draw a ray $B Y$ parallel to $A X$ by making $∠ A B Y$ equal to $∠ B A X$ .

(iv) Mark of three point $A_{1}, A_{2}, A_{3}$ on $A X$ and 4 points $B_{1}, B_{2} m B_{3}$ , $B_{4}$ on $B Y$ such that $A A_{1} = A_{1} A_{2} = A_{2} A_{3}$ $= B B_{1} = B_{1} B_{2} = B_{2} B_{3} = B_{2} B_{4}$ .

(v) Join $A_{3} B_{4}$ . Suppose it intersects $A B$ at a point $P$ .

Then, $P$ is the point dividing $A B$ internally in the ratio 3:4.

CONSTRUCTION

10.3 CONTRUCTION OF A TRIANGLE SIMILAR TO A GIVEN TRIANGLE :

Scale Factor: The ratio of the sides of the triangle to be constructed with the corresponding sides of the given triangle is known as their scale factor.

alt text

CONSTRUCTION

Steps of Construction when $m < n$ :

(i) Construct the given triangle $A B C$ by using the given data.

(ii) Take any one of the three side of the given triangle as base. Let $A B$ be the base of the given triangle.

(iii) At one end, say $A$ , of base $A B$ . Construct an acute angle $∠ B A X$ below the base $A B$ .

(iv) Along $A X$ mark of $n$ points $A_{1}, A_{2}, A_{3}, \dots . . A_{n}$ such that $A A_{1} = A_{1} A_{2} = \dots . . = A_{n - 1} A_{n}$ .

(v) Join $A_{n} B$ .

(vi) Draw $A_{m} B^{'}$ parallel to $A_{n} B$ which meets $A B$ at $B^{'}$ . (vii) From $B^{'}$ draw $B^{'} C^{'} | C B$ meeting $A C$ at $C^{'}$ .

Triangle $A B^{'} C^{'}$ is the required triangle each of whose side is $(\frac{m}{n})^{th}$ of the corresponding side of $△ A B C$ .

CONSTRUCTION

Ex. 3 Construction a $△ A B C$ in which $A B = 5 c m, B C = 6 c m$ and $A C = 7 c m$ . Now, construct a triangle similar to $△ A B C$ such that each of its side is two-third of the corresponding side of $△ A B C$ .

CONSTRUCTION

Sol. Steps of Construction

(i) Draw a line segment $A B = 5 c m$ .

(ii) With $A$ as centre and radius $A C = 7 c m$ , draw an arc.

(iii) With $B$ as centre and $B C = 6 c m$ , draw another arc, intersecting the arc draw in step (ii) at $C$ .

(iv) Join $A C$ and $B C$ to obtain $△ A B C$ .

(v) Below $A B$ , make an acute angle $∠ B A X$ .

(vi) Along $A X$ , mark off three points (greater of 2 and 3 in $\frac{2}{3}$ ) $A_{1}, A_{2}, A_{3}$ such that $A A_{1} = A_{1} A_{2} = A_{2} A_{3}$ .

(vii) Join $A_{3} B$ .

(viii) Draw $A_{2} B^{'} | | A_{3} B$ , meeting $A B$ at $B^{'}$ .

CONSTRUCTION

(iv) From $B^{'}$ , draw $B^{'} C^{'} | | B C$ , meeting $A C$ at $C^{'}$ .

$A B^{'} C^{'}$ is the required triangle, each of the whose sides is two-third of the corresponding sides of $△ A B C$ .

CONSTRUCTION

Steps of Construction when $m > n$ :

(i) Construct the given triangle by using the given data.

(ii) Take any of the three sides of the given triangle and consider it as the base. Let $A B$ be the base of the given triangle.

(iii) At one end, say $A$ , of base $A B$ construct an acute angle $∠ B A X$ below base $A B$ i.e. on the composite side of the vertex C.

(iv) Along $A X$ , mark-off $m$ (large of $m$ and $n$ ) points $A_{1}, A_{2}, \dots . . A_{m}$ on $A X$ such that $A A_{1} = A_{1} A_{2} = \dots . A_{m - 1} A_{m}$ .

(v) Join $A_{n}$ to $B$ and draw a line through $A_{m}$ parallel to $A_{n} B$ , intersecting the extended line segment $A B$ at $B$ ‘.

CONSTRUCTION

(vi) Draw a line through $B^{'}$ parallel to $B C$ intersecting the extended line segment $A C$ at $C^{'}$ .

(vii) $△ A B^{'} C^{'}$ so obtained is the required triangle.

CONSTRUCTION

Ex. 4 Draw a triangle $A B C$ with side $B C = 7 c m, ∠ B = 45^{\circ}, ∠ A = 150^{\circ}$ Construct a triangle whose side are $(4 / 3)$ times the corresponding side of $△ A B C$ .

CONSTRUCTION

Sol. In order to construct $△ A B C$ , follow the following steps :

(i) Draw $B C = 7 c m$ .

(ii) At B construct $∠ C B X = 45^{\circ}$ and at $C$ construct $∠ B C Y = 180^{\circ} - (45^{\circ} + 105^{\circ}) = 30^{\circ}$ Suppose $B C$ and $C Y$ intersect at $A$ . $△ A B C$ so obtained is the given triangle.

(iii) Construct an acute angle $∠ C B Z$ at $B$ on opposite side of vertex $A$ of $△ A B C$ .

CONSTRUCTION

(iv) Mark-off four (greater of 4 and 3 in $\frac{4}{3}$ ) points, $B_{1}, B_{2}, B_{3}, B_{4}$ on $B Z$ such that $B_{2} - B_{1} B_{2} = V_{2} B_{3} = B_{3} B_{4}$ .

(v) Join $B_{3}$ ( the third point) to $C$ and draw a line through $B_{4}$ parallel to $B_{3} C$ , intersecting the extended line segment $B C$ at $C^{'}$ .

(vi) Draw a line through $C^{'}$ parallel to $C A$ intersecting the extended line segment $B A$ at $A^{'}$ Triangle $A^{'} B C$ ’ so obtained is the required triangle such that $\frac{A^{'} B^{'}}{A B} = \frac{B C^{'}}{B C} = \frac{A^{'} C^{'}}{A C} = \frac{4}{3}$

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CONSTRUCTION

10.4 CONSTRCUTION OF TANGENT TO A CIRCLE :

10.4 (a)To Draw the Tangent to a Circle at a Given Point on it, When the Centre of the Circle is Known :

Given: A circle with centre $O$ and a point $P$ and it.

Required : To draw the tangent to the circle at $P$ .

CONSTRUCTION

Steps of Construction.

(i) Join OP.

(ii) Draw a line AB perpendicular to $O P$ at the point $P$ . APB is the required tangent at $P$ .

CONSTRUCTION

Ex. 5 Draw a circle of diameter $6 c m$ with centre O. Draw a diameter AOB. Through A or B draw tangent to the circle.

CONSTRUCTION

Sol. Given : A circle with centre $O$ and a point $P$ on it.

Required : To draw tangent to the circle at $B$ or $A$ .

Steps of Construction.

(i) With $O$ as centre and radius equal to $3 c m$ ( $6 \div 2$ ) draw a circle.

(ii) Draw a diameter $A O B$ .

CONSTRUCTION

(iii) Draw $C D ⊥ A B$ .

(iv) So. CD is the required tangent.

CONSTRUCTION

10.4 (b)To Draw the Tangent to a Circle at a Given Point on it, When the Centre of the Circle is not Known :

Given : A circle and a point $P$ on it.

Required : To draw the tangent to the circle at $P$ .

Steps of Construction

(i) Draw any chord $P Q$ and Joint $P$ and $Q$ to a point $R$ in major arc $P Q$ (or minor arc $P Q$ ).

(ii) Draw $∠ Q P B$ equal to $∠ P R Q$ and on opposite side of chord $P Q$ .

CONSTRUCTION

The line BPA will be a tangent to the circle at $P$ .

CONSTRUCTION

Ex. 6 Draw a circle of radius $4.5 c m$ . Take a point $P$ on it. Construct a tangent at the point $P$ without using the centre of the circle. Write the steps of construction.

CONSTRUCTION

Sol. Given : To draw a tangent to a circle at $P$ .

Steps of Construction

(i) Draw a circle of radius $= 4.5 c m$ .

(ii) Draw a chord $P Q$ , from the given point $P$ on the circle.

(iii) Take a point $R$ on the circle and joint $P R$ and $Q R$ .

(iv) Draw $∠ Q P B = ∠ P R Q$ on the opposite side of the chord $P Q$ .

(v) Produce BP to A. Thus, APB is the required tangent.

CONSTRUCTION

10.4 (c) To Draw the Tangent to a Circle from a Point Outside it (External Point) When its Centre is known

Given : $A$ circle with centre $O$ and a point $P$ outside it.

Required : To construct the tangents to the circle from $P$ .

CONSTRUCTION

Steps of Construction :

(i) Join $O P$ and bisect it. Let $M$ be the mid point of OP.

(ii) Taking $M$ as centre and $M O$ as radius, draw a circle to intersect $C (O, r)$ in two points, say $A$ and $B$

(iii) Join PA and $P B$ . These are the required tangents from $P$ to $C (O, r)$

CONSTRUCTION

Ex. 7 Draw a circle of radius $2.5 c m$ . From a point $P, 6 c m$ apart from the centre of a circle, draw two tangents to the circle.

CONSTRUCTION

Sol. Given: A point $P$ is at a distance of $6 c m$ from the centre of a circle of radius $2.5 c m$

Required : To draw two tangents to the circle from the given point P.

CONSTRUCTION

Steps of Construction :

(i) Draw a circle of radius $2.5 c m$ . Let it centre be $O$ .

(ii) Join OP and bisect it. Let $M$ be mid-point of OP.

(iii) Taking $M$ as centre and $M O$ as radius draw a circle to intersect $C$ in two points, say $A$ and $B$ .

(iv) Join PA and PB. These are the required tangents from $P$ to $C$ .

CONSTRUCTION

10.4 (d) To Draw Tangents to a Circle From a Point Outside it (When its Centre is not Known):

Given : $P$ is a point outside the circle.

Required : To draw tangents from a point $P$ outside the circle.

CONSTRUCTION

Steps of Construction :

(i) Draw a secant $P A B$ to intersect the circle at $A$ and $B$ .

(ii) Produce AP to a point $C$ , such that $P A = P C$ .

(iii) With $B C$ as a diameter, draw a semicircle.

(iv) Draw PD $⊥ C B$ , intersecting the semicircle at $D$ .

(v) Taking PD as radius and $P$ as centre, draw arcs to intersect the circle at $T$ and $T$ ‘.

(iv) Join PT and PT’. Then, PT and PT’ are the required tangents.

CONSTRUCTION

Ex. 8 Draw a circle of radius $3 c m$ . From a point $P$ , outside the circle draw two tangents to the circle without using

the centre of the circle.

Given : A point $P$ is outside the circle of radius $3 c m$ .

Required : To draw two tangents to the circle from the point $P$ , without the use of centre.

CONSTRUCTION

Steps of constructing

(i) Draw a circle of radius $3 c m$ .

(ii) Take a point $P$ outside the circle and draw a secant $P A B$ , intersecting the circle at $A$ and $B$ .

(iii) Produce AP to $C$ such that $A P = C P$ .

(iv) Draw a semicircle, wit $C B$ as a diameter.

(v) Draw PD $⊥ A B$ , intersecting the semi-circle AT D.

(vi) With $P D$ as radius and $P$ as centre draw two arcs to intersect the given circle at $T$ and $T$ ‘.

(vii) Joint PT and PT’. Which are the required tangents.

CONSTRUCTION

DAILY PRATICE PROBLEMS 10

SUBEJCTIVE DPP -10.1

1. Draw a circle of radius $2.5 c m$ . Take a point $P$ on it. Draw a tangent to the circle at the point $P$ .

CONSTRUCTION

2. From a point $P$ on the circle of radius $4 c m$ , draw a tangent to the circle without using the centre. Also, write steps of construction.

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3. Draw a circle of radius $3.5 c m$ . Take a point $P$ on it. Draw a tangent to the circle at the point $P$ , without using the centre of the circle.

CONSTRUCTION

4. Draw a circle of radius $3 c m$ . Take a point $P$ at a distance of $5.6 c m$ from the centre of the circle. From the point $P$ , draw two tangents to the circle.

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5. Draw a circle of radius $4.5 c m$ . Take point $P$ outside the circle. Without using the centre of the circle, draw two tangents to the circle from the point $P$ .

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6. Construct a triangle $A B C$ , similar to a given equilateral triangle $P Q R$ with side $5 c m$ . such that each of its side is 6/7th of the corresponding side of the $△ P Q R$ .

CONSTRUCTION

7. Construct a triangle $A B C$ . similar to a given isosceles triangle $P Q R$ with $Q R = 5 c m, P R = P Q = c m$ , such that each of its side is $5 / 3$ of the corresponding sides of the $△ P Q R$ .

CONSTRUCTION

8. Draw a line segment $A B = 7 c m$ . Divide it externally in the ratio of (i) $3 : 5$ (ii) $5 : 3$

CONSTRUCTION

9. Draw a $△ A B C$ with side $B C = 6 c m, A B = 5 c m$ and $∠ A B C = 60^{\circ}$ . Construct a $△ A B^{'} C^{'}$ similar to $△ A B C$ such that sides of $△ A^{'} C^{'}$ are $\frac{3}{4}$ of the corresponding sides of $△ A B C$ .

[CBSE - 2008]

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knowledge-route Maths10 Cha2

title: “Lata knowledge-route-Class10-Math1-2 Merged.Pdf(1)” type: “reveal” weight: 1

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DAILY PRATICE PROBLEMS 10

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