###### Advertisements

###### Advertisements

Find the ratio in which the line segment joining the points A(3,- 3) and B(- 2, 7) is divided by x-axis. Also find the coordinates of the point of division.

###### Advertisements

#### Solution

Solution:

Point p lies on x axis so it’s ordinate is 0 (Using section formula)

Let the ratio be k: 1 Let the coordinate of the point be P(x , 0) As given A(3,-3) and B(-2,7)

The co-ordinates of the point P(x,y), which divide the line segment joining the points A(x_{1},y_{1}) and B(x_{2},y_{2}), internally in the ratio m_{1}:m_{2} are

`P_x=(mx_2+nx_1)/(m+n)`

`P_y=(my_2+ny_1)/(m+n)`

Hence, A(3,-3) be the co-ordinates (x_{1},y_{1}) and B(-2,7) be the co-ordinates (x_{2},y_{2})

m = k

n = 1

0 = k x 7 + 1 x -3 / (k+1)

0(k+1) =7k -3

0 =7k - 3

3=7k

k = 3 /7

k:1 = 3 : 7

`P_x=(mx_2+nx_1)/(m+n)`

`P_x=[(3/7xx-2)+(1xx3)]/(3/7+1)=2.41`

#### APPEARS IN

#### RELATED QUESTIONS

Write down the equation of a line whose slope is 3/2 and which passes through point P, where P divides the line segment AB joining A(-2, 6) and B(3, -4) in the ratio 2 : 3.

ΔRST ~ ΔUAY, In ΔRST, RS = 6 cm, ∠S = 50°, ST = 7.5 cm. The corresponding sides of ΔRST and ΔUAY are in the ratio 5 : 4. Construct ΔUAY.

Draw a line segment of length 7.6 cm and divide it in the ratio 5:8. Measure the two parts. Give the justification of the construction.

Construct a triangle of sides 4 cm, 5cm and 6cm and then a triangle similar to it whose sides are `2/3` of the corresponding sides of the first triangle. Give the justification of the construction.

Construct an isosceles triangle with base 8 cm and altitude 4 cm. Construct another triangle whose sides are `2/3` times the corresponding sides of the isosceles triangle.

Draw a line segment of length 7 cm and divide it internally in the ratio 2 : 3.

Construct a triangle similar to a given ΔABC such that each of its sides is (5/7)^{th} of the corresponding sides of Δ ABC. It is given that AB = 5 cm, BC = 7 cm and ∠ABC = 50°.

Draw a right triangle ABC in which AC = AB = 4.5 cm and ∠A = 90°. Draw a triangle similar to ΔABC with its sides equal to (5/4)^{th} of the corresponding sides of ΔABC.

Draw a right triangle in which the sides (other than hypotenuse) are of lengths 5 cm and 4 cm. Then construct another triangle whose sides are 5/3^{th} times the corresponding sides of the given triangle.

Construct the circumcircle and incircle of an equilateral ∆XYZ with side 6.5 cm and centre O. Find the ratio of the radii of incircle and circumcircle.

∆PQR ~ ∆LTR. In ∆PQR, PQ = 4.2 cm, QR = 5.4 cm, PR = 4.8 cm. Construct ∆PQR and ∆LTR, such that `"PQ"/"LT" = 3/4`.

∆AMT ~ ∆AHE. In ∆AMT, AM = 6.3 cm, ∠TAM = 50°, AT = 5.6 cm. `"AM"/"AH" = 7/5`. Construct ∆AHE.

∆ABC ~ ∆LBN. In ∆ABC, AB = 5.1 cm, ∠B = 40°, BC = 4.8 cm, \[\frac{AC}{LN} = \frac{4}{7}\]. Construct ∆ABC and ∆LBN.

Find the ratio in which point T(–1, 6)divides the line segment joining the points P(–3, 10) and Q(6, –8).

Find the ratio in which point P(k, 7) divides the segment joining A(8, 9) and B(1, 2). Also find k.

Find the co-ordinates of the points of trisection of the line segment AB with A(2, 7) and B(–4, –8).

If A(–14, –10), B(6, –2) is given, find the coordinates of the points which divide segment AB into four equal parts.

If A (20, 10), B(0, 20) are given, find the coordinates of the points which divide segment AB into five congruent parts.

Given A(4, –3), B(8, 5). Find the coordinates of the point that divides segment AB in the ratio 3 : 1.

Δ SHR ∼ Δ SVU. In Δ SHR, SH = 4.5 cm, HR = 5.2 cm, SR = 5.8 cm and

SHSV = 53 then draw Δ SVU.

Δ AMT ∼ ΔAHE. In Δ AMT, MA = 6.3 cm, ∠MAT = 120°, AT = 4.9 cm, `(MA)/(HA) = 7/5`. construct Δ AHE.

Find the co-ordinates of the centroid of the Δ PQR, whose vertices are P(3, –5), Q(4, 3) and R(11, –4)

**Choose the correct alternative:**

______ number of tangents can be drawn to a circle from the point on the circle.

**Choose the correct alternative:**

In the figure ΔABC ~ ΔADE then the ratio of their corresponding sides is ______

∆ABC ~ ∆PBQ. In ∆ABC, AB = 3 cm, ∠B = 90°, BC = 4 cm. Ratio of the corresponding sides of two triangles is 7: 4. Then construct ∆ABC and ∆PBQ

ΔRHP ~ ΔNED, In ΔNED, NE = 7 cm, ∠D = 30°, ∠N = 20° and `"HP"/"ED" = 4/5`. Then construct ΔRHP and ΔNED

ΔPQR ~ ΔABC. In ΔPQR, PQ = 3.6cm, QR = 4 cm, PR = 4.2 cm. Ratio of the corresponding sides of triangle is 3 : 4, then construct ΔPQR and ΔABC

Construct an equilateral ∆ABC with side 5 cm. ∆ABC ~ ∆LMN, ratio the corresponding sides of triangle is 6 : 7, then construct ΔLMN and ΔABC

ΔAMT ~ ΔAHE. In ΔAMT, AM = 6.3 cm, ∠MAT = 120°, AT = 4.9 cm, `"AM"/"HA" = 7/5`, then construct ΔAMT and ΔAHE

ΔABC ~ ΔPBR, BC = 8 cm, AC = 10 cm , ∠B = 90°, `"BC"/"BR" = 5/4` then construct ∆ABC and ΔPBR

If the point P (6, 7) divides the segment joining A(8, 9) and B(1, 2) in some ratio, find that ratio

**Solution:**

Point P divides segment AB in the ratio m: n.

A(8, 9) = (x_{1}, y_{1}), B(1, 2 ) = (x_{2}, y_{2}) and P(6, 7) = (x, y)

Using Section formula of internal division,

∴ 7 = `("m"(square) - "n"(9))/("m" + "n")`

∴ 7m + 7n = `square` + 9n

∴ 7m – `square` = 9n – `square`

∴ `square` = 2n

∴ `"m"/"n" = square`

To construct a triangle similar to a given ΔABC with its sides `3/7` of the corresponding sides of ΔABC, first draw a ray BX such that ∠CBX 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 and next step is to join ______.

For ∆ABC in which BC = 7.5cm, ∠B =45° and AB - AC = 4, select the correct figure.

Match the following based on the construction of similar triangles, if scale factor `(m/n)` is.

Column I |
Column II |
||

i | >1 | a) | The similar triangle is smaller than the original triangle. |

ii | <1 | b) | The two triangles are congruent triangles. |

iii | =1 | c) | The similar triangle is larger than the original triangle. |

Draw a line segment AB of length 10 cm and divide it internally in the ratio of 2:5 Justify the division of line segment AB.