###### Advertisements

###### Advertisements

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

#### Options

12

-2

−12, −2

12, −2

###### Advertisements

#### Solution

If area of triangle is 35 square units with vertices (2, −6), (5, 4), and (k, 4). Then k is **12, −2**.

**Explanation:**

Given, the vertices of the triangle are (2, -6), (5, 4) and (k, 4);

`Delta` Area of = `Delta = 1/2 abs ((x_1,y_1,1),(x_2,y_2,1),(x_3,y_3,1))`

x_{1} = 2, y_{1} = 6, x_{2} = 5, y_{2} = 4, x_{3} = k, y_{3} = 4

`Delta` Area of `pm 35`

`pm 35 = 1/2 [2(4 - 4) + 6(5 - k) + k (120 - 4 k)`

`=> pm 35 = 1/2 [ 2 xx 0 + 6(5 - k) + 1 (20 - 4 k)]`

`=> pm 70 = 6(5 - k) + 20 - 4 k`

`=> pm 70 = 30 - 6 k + 20 - 4 k`

`=> pm 70 = 50 - 10 k`

`=> pm 70 = 5 - k`

7 = 5 - k

⇒ k = 5 - 7

k = -2

-7 = 5 - k

⇒ - 12 = - k

⇒ k = 12

अत: k = 12, -2

#### APPEARS IN

#### RELATED QUESTIONS

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

Find the area of a triangle whose vertices are A(3, 2), B (11, 8) and C(8, 12).

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

Find the area of the quadrilateral whose vertices, taken in order, are (-4, -2), (-3, -5), (3, -2) and (2, 3).

**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)

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

Find equation of line joining (3, 1) and (9, 3) using determinant.

**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)

Find the centroid of the triangle whosw vertices is (1,4), (-1,1) and (3,2) .

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 ?

Find the area of the blades of thc magnetic compass shown in Fig.. 12.27. (Take √11 = 3.32).

Find the third vertex of a ΔABC if two of its vertices are B(-3,1) and C (0,-2) and its centroid is at the origin

For what value of y, are the points P(1, 4), Q(3,y) and R(-3, 16) are collinear ?

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 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 the area of the following triangle:

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.

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 ______.

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)|` = ______.

If the co-ordinates of the vertices of an equilateral triangle with sides of length ‘a’ are (x_{1}, y_{1}), (x_{2}, y_{2}), (x_{3}, y_{3}), then `|(x_1, y_1, 1),(x_2, y_2, 1),(x_3, y_3, 1)|^2 = (3"a"^4)/4`

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

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.

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 points (0, 5), (0, –9) and (3, 6) are collinear.

A(6, 1), B(8, 2) and C(9, 4) are three vertices of a parallelogram ABCD. If E is the midpoint of DC, find the area of ∆ADE.

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

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 |

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

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

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 ______.

Area of a right-angled triangle is 30 cm^{2}. If its smallest side is 5 cm, then its hypotenuse is ______.

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

Triangles having the same base have equal area.

In the given figure, area of ΔPQR is 20 cm^{2} and area of ΔPQS is 44 cm^{2}. Find the length RS, if PQ is perpendicular to QS and QR is 5 cm.

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.

**Find the missing value:**

Base |
Height |
Area of Triangle |

______ | 31.4 mm | 1256 mm^{2} |

**Find the missing value:**

Base |
Height |
Area of Triangle |

22 cm | ______ | 170.5 cm^{2} |