Class 10 - Arihant - All in one (Math)

Draw a line segment AB=8 cm and divide it internally in the ratio 3:2 and also justify it.

Draw a line segment AB = 6.5 cm and divide it internally in the ratio 3:5 and justify the construction.

Construct a triangle similar to given $∆ A B C$ , where AB = 6 cm, BC = 7 cm and AC = 8 cm, with its sides equal to $\frac{3}{4}$ of the corresponding sides of $∆ A B C$ . Also, justify the construction.

Construct a triangle similar to given $∆ A B C$ whose sides are 6 cm, 7 cm adn 8 cm, with its sides equal to $\frac{5}{3}$ of the corresponding sides of $∆ A B C$ .

Draw a circle with the help of circular solid ring. Construct a pair of tangents from a point P outside the circle.

Draw a circle of radius 2.8 cm. From an external point P, draw tangents to the circle without using the centre of the circle.

Draw a pair of tangents to a circle of radius 3 cm which are inclined to each other at an angle of $45 °$ .

Draw a line segment of length 7.6 cm and divide it in that ratio 5:8. Measure the two parts.
Steps of Construction
(i) Draw a line segment AB=7.6 cm.
(ii) Draw a ray AX, making an acute $∠ B A X$ with AB.
(iii) Mark 5+8=13 points, i.e. $A_{1}, A_{2}, A_{3}, A_{4}, . . . A_{12}, A_{13} o n A X,$ such that
$A A_{1} = A_{1} A_{2} = A_{2} A_{3} = . . . = A_{12} A_{13}$
(iv) Join $A_{13} B$
(v) From $A_{5}$ , draws $A_{5} O ∥ A_{13} B$ which intersect AB at O [by making an angle at $A_{5}$ equal to $∠ A A_{13} B$ ].
Then, O is the point on AB which divides it in the ratio 5:8.
So, AO:OB=5:8

Justification
In $∆ A B A_{13}$ , we have
$A_{5} O ∥ A_{13} B ∴ \frac{A O}{O B} = \frac{A A_{5}}{A_{5} A_{13}} [b y b a s i c p r o p o r t i o n a l t h e o r e m] B y c o n s t r u c t i o n, \frac{A A_{5}}{A_{5} A_{13}} = \frac{5}{8} H e n c e, \frac{A O}{O B} = \frac{5}{8} o r A O : O B = 5 : 8 O n m e a s u r i n g, w e f o u n d t h a t A O = 2.9 c m a n d O B = 4.7 c m$

Construct a triangle of sides 4 cmm 5 cm and 6 cm and then a triangle similar to it whose sides are $\frac{2}{3}$ of the corresponding sides of the first triangle.

10.

Construct a triangle with sides 5 cm, 6 cm and 7 cm and then another triangle whose sides are $\frac{7}{5}$ of the corresponding sides of the first triangle.

11.

Construct an isosceles triangle whose base is 8 cm and altitude 4 cm and then another triangle whose sides are $1 \frac{1}{2}$ times the corresponding sides of the isosceles triangle.

12.

Draw a $∆ A B C$ with sides BC = 6 cm, AB = 5 cm and $∠ A B C = 60 °$ . Then, construct a triangle whose sides are $\frac{3}{4}$ of the corresponding sides of the $∆ A B C$ .

13.

Draw a $∆ A B C$ with side BC = 7 cm, $∠ B = 45 °$ and $∠ A = 105 °$ . Then, construct a triangle whose sides are $\frac{4}{3}$ times the corresponding sides of $∆ A B C$ .

14.

Draw a right angled triangle, in which the sides (other than hypotenuse) are of length 4 cm and 3 cm. Then, construct another triangle whose sides are $\frac{5}{3}$ times the corresponding sides of given triangle.

15.

Draw a circle of radius 6 cm. From a point 10 cm away from its centre, construct the pair of tangents to the circle and measure their lengths.

16.

Construction a tangent to a circle of radius 4 cm from a point on the concentric circle of radius 6 cm and measure its length. Also, verify the measurement by actual calculation.

17.

Draw a circle of radius 3 cm. Take two points P and Q on one of its extended diameter each at a distance of 7 cm from its centre. Draw tangents to the circle from these two points P and Q.

18.

Draw a circle of radius 5 cm. Construct a pair of tangents on it, so that they are inclined at 60 $°$ . Measure the lengths of the two tangents.
or
Draw a pair of tangents to a circle of radius 5 cm which are inclined to each other at an angle of
60 $°$ .

19.

Draw a line segment AB of length 8 cm. Taking A as centre, draw a circle of radius 4 cm and taking B as centre, draw another circle of radius 3 cm. Construct tangents to each circle from the centre of the other circle.

20.

Let ABC be a right angled triangle, in which AB = 6 cm, BC = 8 cm and $∠ B = 90 °$ . BD is the perpendicular from B on AC. The circle through B, C and D is drawn. Construct the tangents from A to this circle.

21.

Draw a circle with the help of a bangle. Take a point outside circle. Construct the pair of tangents from this point to circle.

22.

To divide a line segment AB in the ratio 4 : 7, a ray AX is drawn first such that $∠ B A X$ is an acute angle and then points $A_{1}, A_{2}, A_{3}, . . . .$ are located at equal distances on the ray AX. At which point, B is joined?

23.

To divide a line segment AB in the ratio 2 : 5, first a ray AX is drawn, so that $∠ B A X$ is an acute angle and then at equal distances, how many points are located on the ray AX?

24.

In the given figure, find the raito when P divides AB internally.

25.

To draw a pair of tangents to a circle which are inclined to each other at an angle of $60 °$ , it is required to draw tangents at the end points of those two radii of the circle. Find the angle between them.

26.

Given, a triangle with sides AB = 8 cm. To get a line segment AB' = $\frac{3}{4}$ of AB, at what ratio the line segment AB should be divided?

27.

A $∆ A B C$ with sides AB = 4 cm, BC = 5 cm and CA = 7 cm is given. To, construct a triangle whose sides are $\frac{5}{7}$ times of the corresponding sides of the given triangle, first draw a ray BX such that $∠ C B X$ is an acute angle and X lies on the opposite side of A with respect to BC. Then, locate points $B_{1}, B_{2}, B_{3}, . . . . .$ on BX at equal distances. What is the next step to join?

28.

To construct a triangle similar to a given $∆ A B C$ with its sides 8/5 times of the corresponding sides of $∆ A B C$ , draw a ray BX such that $∠ C B X$ is an acute angle and X is on the opposite side of A with respect to BC. How many minimum number of points to be located at equal distances on ray BX?

29.

PQ is a line segment of length 6.4 cm. Geometrically, obtaine point R on PQ such that $\frac{Q R}{P Q} = \frac{5}{8}$ .

30.

Draw two tangents at the end points of the diameter of a circle of radius 3.5 cm. Are these tangents parallel?

31.

Draw a circle of radius 6 cm and draw a tangent to this circle making an angle of 30 $°$ with a line passing through the centre.

32.

Construct a $∆ A B C$ , in which AB = 5 cm, $∠ B = 60 °$ and the altitude CD = 3 cm. Then, construct another trianle whose sides are $\frac{4}{5}$ times of the corresponding sides of $∆ A B C$ .

33.

Draw a $∆ A B C$ which BC = 7 cm, $∠ B = 45 °$ and $∠ C = 60 °$ . Then, construct another trianle, whose sides are $\frac{3}{5}$ times of the corresponding sides of $∆ A B C$ and justify your construction.

34.

Construct an isoceles $∆ A B C$ with base BC = 6 cm, AB = AC and $∠ A = 90 °$ . Draw another similar triangle whose sides are $\frac{4}{5}$ times of the sides of $∆ A B C$ . Justify your construction.

35.

Draw a $∆ A B C$ with side BC = 7 cm, $∠ A B C$ = 60 $°$ and AB = 6 cm. Then, construct another triangle whose sides are $\frac{3}{4}$ times of the corresponding sides of $∆ A B C .$ Justify your construction.

36.

Sanjeev have a piece of cloth of 8 m long. He decided to divide this piece in two persons A and B internally in the ratio 3:4.
(i) Draw a construction of above problem.
(ii) If Sanjeev give 4th part of the piece of the person A, then what value is violated by Sanjeev?

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