Draw a ABC , right-angled at B such that AB = 3 cm and BC = 4cm. Now, Construct a triangle
Construct an isosceles triangle whose base is 9 cm and altitude 5cm. Construct another
Construct a ABC in which BC = 5cm, C 60 and altitude from A equal to 3 cm. Construct
Construct a ABC Sol: in which B= 6.5 cm, AB = 4.5 cm and ABC 60
Draw a line segment AB of length 6.5 cm and divided it in the ratio 4 : 7. Measure each of the two parts.
Draw a line segment AB of length 5.4 cm. Divide it into six equal parts. Write the steps of construction.
Construct a tangent to a circle of radius 4 cm form a point on the concentric circle of radius 6 cm and measure its length. Also, verify the measurement by actual calculation.
Draw a circle of radius 32 cm. Draw a tangent to the circle making an angle 30 with a line passing through the centre.
Write the steps of construction for drawing a pair of tangents to a circle of radius 3 cm , which are inclined to each other at an angle of 60 .
Draw a circle of radius 4.2. Draw a pair of tangents to this circle inclined to each other at an angle of 45
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 form the centre of the other circle.
Draw a circle with the help of a bangle. Take any point P outside the circle. Construct the pair of tangents form the point P to the circle
Draw a circle with center O and radius 4 cm. Draw any diameter AB of this circle. Construct tangents to the circle at each of the two end points of the diameter AB.
2. Draw two tangents to a circle of radius 3.5 cm form a point P at a distance of 6.2 cm form its centre.
Draw a circle of radius 3 cm. Form a point P, 7 cm away from the centre of the circle, draw two tangents to the circle. Also, measure the lengths of the tangents.
Draw a right triangle in which the sides (other than hypotenuse) are of lengths 4 cm and 3
Construct an isosceles triangles whose base is 8 cm and altitude 4 cm and then another
To construct a triangle similar to
7. Construct a ABC in which BC = 8 cm, B 45 and C 60 . Construct another
6. Construct a ABC in which AB = 6 cm, A 30and AB 60 . Construct
Sol: Steps of Construction with base AB’ = 8 cm. Step 1: Draw a line segment AB = 6cm. Step 2: At A, draw ÐXAB = 30°. Step 3: At B, draw ÐYBA = 60°. Suppose AX and BY intersect at C. Thus, DABC is...
5. Construct a ABC with BC = 7 cm, B 60 and AB = 6 cm. Construct another triangle
Construct a triangle with sides 5 cm, 6 cm, and 7 cm and then another triangle whose sides
3. Construct a PQR , in which PQ = 6 cm, QR = 7 cm and PR =- 8 cm. Then, construct
Differentiate the following functions with respect to x: x
Differentiate the following functions with respect to x: . Sin (3x + 5)
Given Sin (3x + 5)
Find the shortest distance between the lines whose vector equations are and
Solution: It is known to us that shortest distance between two lines $\vec{r}=\overrightarrow{a_{1}}+\lambda \overrightarrow{b_{1}}$ and $\vec{r}=\overrightarrow{a_{2}}+\mu \overrightarrow{b_{2}}$...
Find the shortest distance between the lines
Solution: It is known to us that the shortest distance between two lines $\overrightarrow{\mathrm{r}}=\overrightarrow{\mathrm{a}_{1}}+\lambda \overrightarrow{\mathrm{b}_{1}}$ and...
Find the angle between the following pairs of lines:
(i) and
(ii) and
Solution: Let's consider $\theta$ be the angle between the given lines. If $\theta$ is the acute angle between $\vec{r}=\overrightarrow{a_{1}}+\lambda \overrightarrow{b_{1}}$ and...
The Cartesian equation of a line is Write its vector form.
Solution: It is given that The Cartesian equation is $\frac{x-5}{3}=\frac{y+4}{7}=\frac{z-6}{2} \ldots \text { (1) }$ It is known to us that The Cartesian eq. of a line passing through a point...
Find the equation of the line in vector and in Cartesian form that passes through the point with position vector and . is in the direction
Solution: Given: Vector equation of a line that passes through a given point whose position vector is $\vec{a}$ and parallel to a given vector $\vec{b}$ is $\vec{r}=\vec{a}+\lambda \vec{b}$ Let,...
Show that the three lines with direction cosines Are mutually perpendicular.
Solution: Consider the direction cosines of $L_{1}, L_{2}$ and $L_{3}$ be $l_{1}, m_{1}, n_{1} ; l_{2}, m_{2}, n_{2}$ and $l_{3}, m_{3}, n_{3}$. It is known that If $\mathrm{f}_{1}, \mathrm{~m}_{1},...
Find the direction cosines of the sides of the triangle whose vertices are (3, 5, –4), (-1, 1, 2) and (–5, –5, –2).
Solution: Given that, The vertices are $(3,5,-4),(-1,1,2)$ and $(-5,-5,-2)$. Firstly find the direction ratios of $\mathrm{AB}$ Where, $A=(3,5,-4)$ and $B=(-1,1,2)$ Ratio of $A...
Show that the points (2, 3, 4), (–1, –2, 1), (5, 8, 7) are collinear.
Solution: If the direction ratios of two lines segments are proportional, then the lines are collinear. It is given that $\mathrm{A}(2,3,4), \mathrm{B}(-1,-2,1), \mathrm{C}(5,8,7)$ The direction...
If a line has the direction ratios –18, 12, –4, then what are its direction cosines?
Solution: Given that, The direction ratios are $-18,12,-4$ Where, $a=-18, b=12, c=-4$ Consider the direction ratios of the line as $\mathrm{a}, \mathrm{b}$ and $\mathrm{c}$ Direction cosines are...
Find the direction cosines of a line which makes equal angles with the coordinate axes.
Solution: Given that, Angles are equal. Let the angles be $\alpha, \beta, \mathrm{Y}$ The direction cosines of the line be I, $\mathrm{m}$ and $\mathrm{n}$ $I=\cos \alpha, m=\cos \beta \text { and }...
If a line makes angles 90°, 135°, 45° with the x, y and z-axes respectively, find its direction cosines.
Solution: Let's consider the direction cosines of the line be $I, m$ and $n$. Let $\alpha=90^{\circ}, \beta=135^{\circ}$ and $\mathrm{y}=45^{\circ}$ Therefore, $I=\cos \alpha, m=\cos \beta \text {...
In each of the following, give the justification of the construction also:
Draw a circle with the help of a bangle. Take a point outside the circle. Construct the pair of tangents from this point to the circle. Construction Procedure: On the given circle the required...
In each of the following, give the justification of the construction also:
Let ABC be a right 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, D is drawn. Construct the tangents from A to this circle....
In each of the following, give the justification of the construction also:
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...
In each of the following, give the justification of the construction also:
Draw a pair of tangents to a circle of radius 5 cm which are inclined to each other at an angle of 60° Construction Procedure: The given circle tangents can be constructed in the following manner:...
In each of the following, give the justification of the construction also:
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...
In each of the following, give the justification of the construction also:
Construct 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. Construction Procedure:...
In each of the following, give the justification of the construction also:
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. Construction Procedure: The instruction to construct a...
In each of the following, give the justification of the construction also:
Draw a right triangle in which the sides (other than hypotenuse) are of lengths 4 cm and 3 cm. Then construct another triangle whose sides are 5/3 times the corresponding sides of the given...
In each of the following, give the justification of the construction also:
Draw a triangle ABC with side BC = 7 cm, ∠ B = 45°, ∠ A = 105°. Then, construct a triangle whose sides are 4/3 times the corresponding sides of ∆ ABC. We need to find ∠C: Provided that: ∠B = 45°, ∠A...
In each of the following, give the justification of the construction also:
Draw a triangle ABC with side BC = 6 cm, AB = 5 cm and ∠ABC = 60°. Then construct a triangle whose sides are 3/4 of the corresponding sides of the triangle ABC. Construction Procedure: 1. Draw an...
In each of the following, give the justification of the construction also:
Construct an isosceles triangle whose base is 8cm and altitude 4 cm and then another triangle whose sides are times the corresponding sides of the isosceles triangle. Construction Procedure: 1....
In each of the following, give the justification of the construction also:
Construct a triangle with sides 5 cm, 6 cm and 7 cm and then another triangle whose sides are 7/5 of the corresponding sides of the first triangle Construction Procedure: 1. Construct AB a line...
In each of the following, give the justification of the construction also:
Construct a triangle of sides 4 cm, 5 cm and 6 cm and then a triangle similar to it whose sides are 2/3 of the corresponding sides of the first triangle. Construction Procedure: 1. Create a line...
In each of the following, give the justification of the construction also:
Draw a line segment of length 7.6 cm and divide it in the ratio 5 : 8. Measure the two parts.
Construction Procedure: A line segment of length 7.6 cm is divided in 5:8 ratio as follows: 1. Draw a line segment AB with a length of 7.6 cm. 2. Draw a ray AX that intersects line segment AB at an...