###### Advertisements

###### Advertisements

Find a relation between x and y if the points (x, y), (1, 2) and (7, 0) are collinear.

###### Advertisements

#### Solution

If the given points are collinear, then the area of triangle formed by these points will be 0.

Area of Triangle = `1/2 {x_1(y_2-y_3)+x_2(y_3-y_1)+x_3(y_1-y_2)}`

Area = `1/2[x(2-0)+1(0-y)+7(y-2)]`

`0=1/2[2x-y+7y-14]`

`0=1/2[2x+6y-14]`

2x+6y-14=0

x+3y-7 =0

This is the required relation between* x *and *y*.

#### APPEARS IN

#### RELATED QUESTIONS

Find the area of the quadrilateral ABCD whose vertices are respectively A(1, 1), B(7, –3), C(12, 2) and D(7, 21).

In each of the following find the value of '*k*', for which the points are collinear.

(7, -2), (5, 1), (3, -*k*)

The class X students of a secondary school in Krishinagar have been allotted a rectangular plot of land for their gardening activity. Saplings of Gulmohar are planted on the boundary at a distance of 1 m from each other. There is a triangular grassy lawn in the plot as shown in the following figure. The students are to sow seeds of flowering plants on the remaining area of the plot.

(i) Taking A as origin, find the coordinates of the vertices of the triangle.

(ii) What will be the coordinates of the vertices of Δ PQR if C is the origin?

Also calculate the areas of the triangles in these cases. What do you observe?

The vertices of a ΔABC are A (4, 6), B (1, 5) and C (7, 2). A line is drawn to intersect sides AB and AC at D and E respectively, such that `(AD)/(AB) = (AE)/(AC) = 1/4`Calculate the area of the ΔADE and compare it with the area of ΔABC. (Recall Converse of basic proportionality theorem and Theorem 6.6 related to ratio of areas of two similar triangles)

**Find the area of a triangle with vertices at the point given in the following:**

(1, 0), (6, 0), (4, 3)

**Find the area of a triangle with vertices at the point given in the following:**

(2, 7), (1, 1), (10, 8)

**Find the area of a triangle with vertices at the point given in the following:**

(−2, −3), (3, 2), (−1, −8)

Show that points A (a, b + c), B (b, c + a), C (c, a + b) are collinear.

Find values of k if area of triangle is 4 square units and vertices are (−2, 0), (0, 4), (0, k)

Find equation of line joining (1, 2) and (3, 6) using the determinant.

If area of triangle is 35 square units with vertices (2, −6), (5, 4), and (k, 4), then k is ______.

**Find the area of the following triangle:**

Prove that the points (a, 0), (0, b) and (1, 1) are collinear if `1/a+1/b=1`

Find the area of a triangle whose sides are 9 cm, 12 cm and 15 cm ?

Find the area of a triangle two sides of which are 18 cm and 10 cm and the perimeter is 42cm ?

In a ΔABC, AB = 15 cm, BC = 13 cm and AC = 14 cm. Find the area of ΔABC and hence its altitude on AC ?

The perimeter of a triangular field is 540 m and its sides are in the ratio 25 : 17 : 12. Find the area of the triangle ?

prove that the points A (7, 10), B(-2, 5) and C(3, -4) are the vertices of an isosceles right triangle.

Show that the points A (3,1) , B (0,-2) , C(1,1) and D (4,4) are the vertices of parallelogram ABCD.

Find the area of ΔABC whose vertices are:

A( 3,8) , B(-4,2) and C( 5, -1)

If the points P(-3, 9), Q(a, b) and R(4, -5) are collinear and a+b=1, find the value of a and b.

Show that ∆ ABC with vertices A (–2, 0), B (0, 2) and C (2, 0) is similar to ∆ DEF with vertices D (–4, 0), F (4, 0) and E (0, 4) ?

Using determinants, find the values of k, if the area of triangle with vertices (–2, 0), (0, 4) and (0, k) is 4 square units.

In ☐ ABCD, l(AB) = 13 cm, l(DC) = 9 cm, l(AD) = 8 cm, find the area of ☐ ABCD.

Using integration, find the area of the triangle whose vertices are (2, 3), (3, 5) and (4, 4).

Using integration, find the area of triangle ABC, whose vertices are A(2, 5), B(4, 7) and C(6, 2).

What is the area of a triangle with base 4.8 cm and height 3.6 cm?

In ∆PQR, PR = 8 cm, QR = 4 cm and PL = 5 cm.

Find:

(i) the area of the ∆PQR

(ii) QM.

If the sides of a triangle are 3 cm, 4 cm and 5 cm, then the area is

The table given below contains some measures of the right angled triangle. Find the unknown values.

Base |
Height |
Area |

20 cm | 40 cm | ? |

The table given below contains some measures of the right angled triangle. Find the unknown values.

Base |
Height |
Area |

5 feet | ? | 20 sq.feet |

The table given below contains some measures of the right angled triangle. Find the unknown values.

Base |
Height |
Area |

? | 12 m | 24 sq.m |

A field is in the shape of a right angled triangle whose base is 25 m and height 20 m. Find the cost of levelling the field at the rate of ₹ 45 per sq.m^{2}

If Δ = `|(1, x, x^2),(1, y, y^2),(1, z, z^2)|`, Δ_{1} = `|(1, 1, 1),(yz, zx, xy),(x, y, z)|`, then prove that ∆ + ∆_{1} = 0.

In a triangle ABC, if `|(1, 1, 1),(1 + sin"A", 1 + sin"B", 1 + sin"C"),(sin"A" + sin^2"A", sin"B" + sin^2"B", sin"C" + sin^2"C")|` = 0, then prove that ∆ABC is an isoceles triangle.

If A, B, C are the angles of a triangle, then ∆ = `|(sin^2"A", cot"A", 1),(sin^2"B", cot"B", 1),(sin^2"C", cot"C", 1)|` = ______.

Show that the points (a + 5, a – 4), (a – 2, a + 3) and (a, a) do not lie on a straight line for any value of a.

The area of a triangle with vertices (–3, 0), (3, 0) and (0, k) is 9 sq.units. The value of k will be ______.

The value of the determinant `abs((1,"x","x"^3),(1,"y","y"^3),(1,"z","z"^3))` is ____________.

If the points (3, -2), (x, 2), (8, 8) are collinear, then find the value of x.

Find the area of the triangle whose vertices are (-2, 6), (3, -6), and (1, 5).

Let `Delta = abs (("x", "y", "z"),("x"^2, "y"^2, "z"^2),("x"^3, "y"^3, "z"^3)),` then the value of `Delta` is ____________.

The area of a triangle with base 4 cm and height 6 cm is 24 cm^{2}.

The base and the corresponding altitude of a parallelogram are 10 cm and 3.5 cm, respectively. The area of the parallelogram is 30 cm^{2}.

Find the cost of laying grass in a triangular field of sides 50 m, 65 m and 65 m at the rate of Rs 7 per m^{2}.

Find the area of the trapezium PQRS with height PQ given in figure

The area of a trapezium is 475 cm^{2} and the height is 19 cm. Find the lengths of its two parallel sides if one side is 4 cm greater than the other.

A rectangular plot is given for constructing a house, having a measurement of 40 m long and 15 m in the front. According to the laws, a minimum of 3 m, wide space should be left in the front and back each and 2 m wide space on each of other sides. Find the largest area where house can be constructed.

**Find the missing value:**

Base |
Height |
Area of parallelogram |

______ | 15 cm | 154.5 cm^{2} |

**Find the missing value:**

Base |
Height |
Area of parallelogram |

______ | 8.4 cm | 48.72 cm^{2} |

The area of a triangle with vertices A, B, C is given by ______.

Area of triangle MNO in the figure is ______.

In the given figure, if PR = 12 cm, QR = 6 cm and PL = 8 cm, then QM is ______.

In the given figure, ΔMNO is a right-angled triangle. Its legs are 6 cm and 8 cm long. Length of perpendicular NP on the side MO is ______.

Observe all the four triangles FAB, EAB, DAB and CAB as shown in the given figure.

- All triangles have the same base and the same altitude.
- All triangles are congruent.
- All triangles are equal in area.
- All triangles may not have the same perimeter.

Ratio of the area of ∆WXY to the area of ∆WZY is 3:4 in the given figure. If the area of ∆WXZ is 56 cm^{2} and WY = 8 cm, find the lengths of XY and YZ.

In the given figure, triangle AEC is right-angled at E, B is a point on EC, BD is the altitude of triangle ABC, AC = 25 cm, BC = 7 cm and AE = 15 cm. Find the area of triangle ABC and the length of DB.

Let a vector `αhati + βhatj` be obtained by rotating the vector `sqrt(3)hati + hatj` by an angle 45° about the origin in counter-clockwise direction in the first quadrant. Then the area of triangle having vertices (α, β), (0, β) and (0, 0) is equal to ______.

If (a, b), (c, d) and (e, f) are the vertices of ΔABC and Δ denotes the area of ΔABC, then `|(a, c, e),(b, d, f),(1, 1, 1)|^2` is equal to ______.

Using determinants, find the area of ΔPQR with vertices P(3, 1), Q(9, 3) and R(5, 7). Also, find the equation of line PQ using determinants.