Area of a Triangle Using Coordinate of Vertices

Q. The area of the triangle formed by the lines x = 3, y = 4 and x = y is

(a) ½ sq. unit
(b) 1 sq. unit
(c) 2 sq. unit
(d) None of these

Q. The area of the triangle formed by the lines y = x, x = 6 and y = 0 is

(a) 36 sq. units
(b) 18 sq. units
(c) 9 sq. units
(d) 72 sq. units

Q.

For what value of x will the points (x, – 1), (2, 1) and (4, 5) lie on a line?

Q.

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 lawn in the plot as shown in the Fig. The students are to sow the seeds of flowering plants on the remaining area of the plot. $\left(i\right)$ Taking A as origin, find the coordinates of the vertices of the triangle. $\left(ii\right)$ What will be the coordinates of the vertices of triangle PQR if C is the origin? Also calculate the areas of the triangles in these cases. What do you observe?

Q.

The area of the triangle formed by the line $\frac{x}{a}+\frac{y}{b}=1$ with the coordinate axes is

(a) ab
(b) 2ab
(c) $\frac{1}{2}ab$
(d) $\frac{1}{4}ab$

Q. Show that the point P (−4, 2) lies on the line segment joining the points A (−4, 6) and B (−4, −6).

Q. The area of the triangle formed by (a, b + c), (b, c + a) and (c, a + b)

(a) a + b + c

(b) abc

(c) (a + b + c)²

(d) 0

Q. The area of a triangle is 5. Two of its vertices are (2, 1) and (3, −2). The third vertex lies on y = x + 3. Find the third vertex.

Q. Write the condition of collinearity of points (x1, y1), (x2, y2) and (x3, y3).

Q. If the point P (m, 3) lies on the line segment joining the points $A\left(-\frac{2}{5},6\right)$and B (2, 8), find the value of m.

Q. The area of the triangle formed by the lines 2x + 3y = 12, x − y − 1 = 0 and x = 0 (as shown in Fig. 3.23), is

(a) 7 sq. units
(b) 7.5 sq. units
(c) 6.5 sq. units
(d) 6 sq. units

Q.

If A(-3, 5) , B (-2, -7) , C(1, -8) and D(6, 3) are the vertices of a quadrilateral ABCD find its area.

Q. Find the value of k if points A(k, 3), B(6, −2) and C(−3, 4) are collinear.

Q. Write the formula for the area of the triangle having its vertices at (x₁, y₁), (x2, y₂) and (x₃, y₃).

Q.

The area of a quadrilateral whose vertices taken in order are (–4, –2), (–3, –5), (3, –2) and (2, 3) is _______.

1. 26 sq. units

2. 30 sq. units

3. 28 sq. units

4. 27 sq. units

Q. The vertices of a triangle are (𝟐, 𝟑), (−𝟏, 𝟎) and (𝟐, −𝟒). Find the area of the triangle.

Q. Prove that the points (a, 0), (0, b) and (1, 1) are collinear if $\frac{1}{a}+\frac{1}{b}=1$.

Q. If points A(-2, 1), B(a, b), and C(4, -1) are colllinear and a-b=1. Find the values of a and b.

Q.

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

Q.

The area of a triangle having vertices (a, b+ c), (b, c + a) and (c, a + b) is ___.

1. 0

2. 1

3. a + b + c

4. $a^2$ + $b^2$ + $c^2$

Q. A triangle is formed by joining three points whose coordinates are integers. If the x-unit and the y-unit are each 1 cm, then the area of the triangle, in sq. cm:

1. must be an integer
2. may be irrational
3. must be irrational
4. must be rational
5. will be an integer only if the triangle is equilateral

Q. Solve this:

Q. If the area of the quadrilateral whose angular points taken in order are (1, 2), (-5, 6), (7, -6), (h, -2) be zero. Show that h = 3.

Q.

The area of a triangle having vertices (a, b+ c), (b, c + a) and (c, a + b) is ___.

1. 0

2. 1

3. a + b + c

4. $a^2$ + $b^2$ + $c^2$

Q. If 3x – 4y + 12 = 0 and 3x + 2y – 6 = 0, then area of triangle formed by the given lines and x-axis is (in sq. units)

Q.

The area of a triangle is 5 square units. Two of its vertices are (2, 1) and (3, -2) and the third vertex lies on y = x + 3, the third vertex is

1. $(3,4)$

2. $(\frac{5}{2}$, $\frac{13}{2})$

3. $(\frac{7}{2}$, $\frac{13}{2})$

4. $(\frac{-3}{2}$, $\frac{3}{2})$

Q.

A(a, b), B(c, d) and C(e, f) are the vertices of a triangle.
i) AB + BC > AC
ii) Area of the triangle = $\left(\frac{1}{2}\right)$[a(c-e)+c(f-b)+e(b-d)]
Which of the following are true?

1. Only (i) is true

2. Only (ii) is true

3. Both (i) and (ii) is true

4. Neither (i) nor (ii) is true

Q.

The points A(-a, b), B(-a, c) and C(a, c) form the vertices of a triangle. Which of the following is true?

1. $\Delta ABC$ is a right angled triangle, right angled at A.
2. Area of the triangle = b(a - c)
3. Area of the triangle = a(c - b)
4. The triangle is equilateral.

Q.

Find the area of the triangle with coordinates A(-2, 1), B(3, 3) and C(1, -2).

1. 10 sq. units

2. 10.5 sq. units

3. 11 sq. units

4. 11.5 sq. units

Q. The area of a triangle is 5. Two of its vertices are (2, 1) and (3, −2). The third vertex lies on y = x + 3. Find the third vertex.

Q. What is the area of the right-angled triangle $\Delta MNO$, right angled at N.

1. $\frac{1}{2}\times MN\times NO$
2. $\frac{1}{2}\times MN\times MO$
3. $\frac{1}{2}\times MO\times NO$
4. $MN\times NO$