###### Advertisements

###### Advertisements

Find the values of k so that the area of the triangle with vertices (k + 1, 1), (4, -3) and (7, -k) is 6 sq. units.

###### Advertisements

#### Solution

Let A(k + 1, 1), B(4,-3) and c(7,-k) are the vertices of the triangle.

Given that the area of the triangle is 6sq. units.

Area of the triangle is given by

`A=1/2[x_1(y_2-y_3)+x_2(y_3-y_1)+x_3(y_1-y_2)]`

`6=1/2[x_1(y_2-y_3)+x_2(y_3-y_1)+x_3(y_1-y_2)]`

`12=(k+1)(-3+k)+4(-k-1)+7(1+3)`

12=−3k+

`12=-3k+k^2-3+k-4k-4+28`

`12=k^2-6k+21`

`k^2-6k+21-12=0`

`k^2-6k+9=0`

`(k-3)^2=0`

`k=3,3`

#### APPEARS IN

#### RELATED QUESTIONS

Find the area of the triangle formed by joining the mid-point of the sides of the triangle whose vertices are (0, –1), (2, 1) and (0, 3). Find the ratio of area of the triangle formed to the area of the given triangle.

The vertices of ∆ABC = 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 `\frac{AD}{AB}=\frac{AE}{AC}=\frac{1}{4}` .Calculate the area of ∆ADE and compare it with the area of ∆ABC

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

Let A (4, 2), B (6, 5) and C (1, 4) be the vertices of ΔABC.

(i) The median from A meets BC at D. Find the coordinates of point D.

(ii) Find the coordinates of the point P on AD such that AP: PD = 2:1

(iii) Find the coordinates of point Q and R on medians BE and CF respectively such that BQ: QE = 2:1 and CR: RF = 2:1.

(iv) What do you observe?

(v) If A(x_{1}, y_{1}), B(x_{2}, y_{2}), and C(x_{3}, y_{3}) are the vertices of ΔABC, find the coordinates of the centroid of the triangle.

**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 (k, 0), (4, 0), (0, 2)

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:**

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

Find the area of the quadrilaterals, the coordinates of whose vertices are

(−3, 2), (5, 4), (7, −6) and (−5, −4)

Show that the following sets of points are collinear.

(1, −1), (2, 1) and (4, 5)

Prove that the points (2,3), (-4, -6) and (1, 3/2) do not form a triangle.

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 ?

Show that the following points are collinear:

A(-5,1), B(5, 5) and C(10, 7)

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?

Find BC, if the area of the triangle ABC is 36 cm^{2} and the height AD is 3 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}

Let ∆ = `|("A"x, x^2, 1),("B"y, y^2, 1),("C"z, z^2, 1)|`and ∆_{1} = `|("A", "B", "C"),(x, y, z),(zy, zx, xy)|`, then ______.

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.

Show that the ∆ABC is an isosceles triangle if the determinant

Δ = `[(1, 1, 1),(1 + cos"A", 1 + cos"B", 1 + cos"C"),(cos^2"A" + cos"A", cos^2"B" + cos"B", cos^2"C" + cos"C")]` = 0

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.

If the points (a_{1}, b_{1}), (a_{2}, b_{2}) and(a_{1} + a_{2}, b_{1} + b_{2}) are collinear, then ____________.

If the points (2, -3), (k, -1), and (0, 4) are collinear, then find the value of 4k.

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 the triangle ABC with the vertices A(-5, 7), B(-4, -5) and C(4, 5) is ______.

If the points A(1, 2), O(0, 0) and C(a, b) are collinear, then ______.

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

The area of ∆ABC is 8 cm^{2} in which AB = AC = 4 cm and ∠A = 90º.

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

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.

The dimensions of a rectangle ABCD are 51 cm × 25 cm. A trapezium PQCD with its parallel sides QC and PD in the ratio 9:8, is cut off from the rectangle as shown in the figure. If the area of the trapezium PQCD is `5/6` h part of the area of the rectangle, find the lengths QC and PD.

**Find the missing value:**

Base |
Height |
Area of parallelogram |

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

Area of triangle MNO in the figure is ______.

Area of triangle PQR is 100 cm^{2} as shown in the below figure. If altitude QT is 10 cm, then its base PR 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 ______.

Area of a triangle = `1/2` base × ______.

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.