Area of a Triangle Given Its Vertices

Q.

Question 90

Solve the following:

Two equal sides of a triangle are each 4 m less than three times the third side. Find the dimensions of the triangle, if its perimeter is 55m.

Q.

A clock strikes once at 1 o'clock, twice at 2 o'clock, thrice at 3 o’clock and so on. How many times will it strike in 12 hours?

1. 156

2. 66

3. 78

4. 132

Q.

Draw the graphs of the equations $5x-y=5$ and $3x-y=3$. Determine the co-ordinates of the vertices of the triangle formed by these lines and the $y$ axis.

Q.

If the centroid of a triangle is (1, 4) and two of its vertices are (4, -3) and (-9, 7), then the area of the triangle is

1. 183 sq. units

2. $\frac{183}{2}sq.units.$

3. 366 sq. units

4. $\frac{183}{4}sq.units.$

Q.

The value of k for which the points $A(2,3),B(4,k),$ and $C(6,-3)$ are collinear.

1. $k=9$

2. $k=-6$

3. $k=0$

4. $k=4$

Q.

If the area of the triangle formed by the points$(x,2x),(-2,6)$ and$(3,1)$ is $5$ square units , then $x$
(a)$23$
(b) $35$
(c)$3$
(d) $5$

Q.

Question 6
The vertices of triangle 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 triangle ADE and compare it with area of triangle ABC.

Q.

Show that the points$O(0,0),A(3,\sqrt{3})andB(3,-\sqrt{3})$ are the vertices of an equilateral triangle. Find the area of this triangle.

Q.

Question 5
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 figure. The students are to sow the seeds of flowering plants on the remaining area of the plot.
(a) Taking A as origin, find the coordinates of the vertices of the triangle.
(b) What will be the coordinates of the vertices of triangle PQR if C is the origin.
(c) Also calculate the areas of the triangles in these cases. What do you observe?

Q.

Find the area of the triangle formed by joining the midpoints of the sides of the triangle whose vertices are A(2, 1), B(4, 3) and C(2, 5)

Q. Find the area of quadrilateral ABCD whose vertices are A(−5, 7), B(−4, −5), C(−1, −6) and D(4, 5).

Q.

Draw the graph of $y=x^2+x-12$ and hence solve $x^2+2x+2=0$.

Q.

If the points A(1, -2) , B(2, 3) , C(-3, 2) and D(-4, -3) are the vertices of a parallelogram ABCD, then taking AB as the base, find the height of the parallelogram.

Q.

Find the area of a quadrilateral $ABCD$, the coordinates of whose vertices are $A(-3,2),B(5,4),C(7,-6)$ and $D(-5,-4)$.

Q. The vertices of a $\Delta OBC$ are O(0, 0), B(–3, –1) and C(–1, –3). The equation of a line parallel to BC and intersecting sides OB and OC whose distance from the origin is $\frac{1}{2}$, is

1. 
2. 
3. 
4. 

Q. If the points A(x, y), B(3, 6) and C(−3, 4) are collinear, show that x − 3y + 15 = 0

Q.

The are of $△ABC$ with vertices A(a, 0), O(0, 0) and B(0, b) in square units is

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

Q.

The area of triangle ABC with A(3, 2), B(11, 8) and C(8, 12) in square units is

1. 25

2. 20

3. 35

4. 30

Q.

Find the area of $△ABC$ whose vertices are :

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

(ii) A (-5, 7), B(-4, -5) and C(4, 5)

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

(iv) A (10, -6), B (2, 5) and C(-1, 3)

Q.

Find the area of quadrilateral PQRS whose vertices are P(-5, -3), Q(-4, -6), R(2, -3) and S(1, 2).

Q.

The area of the triangle with vertices (a, b+ c), (b, c + a) and (c, a + b) is

1. a² + b² + c²

2. 0

3. 1

4. a + b + c

Q.

In each of the following find the value of ‘k’, for which the points are collinear.

(i) (7, − 2), (5, 1), (3, − k) (ii) (8, 1), (k, − 4), (2, − 5)

Q. Write the perimeter of the triangle formed by the points O (0, 0), A (a, 0) and B (0, b).

Q.

If A(-5, 7), B(-4, -5), C(-1, -6) and D(4, 5) are the vertices of a quadrilateral then its area is

1. 80 sq.units

2. 50 sq.units

3. 72 sq. units

4. 27 sq.units

Q.

The area of the triangle formed by the lines $\mathrm{x}=0$, $\mathrm{y}=0$ and $\frac{\mathrm{x}}{\mathrm{a}}+\frac{\mathrm{y}}{\mathrm{b}}=1$ is

1. $\mathrm{ab}$

2. $\frac{\mathrm{ab}}{2}$

3. $2\mathrm{ab}$

4. $\frac{\mathrm{ab}}{3}$

Q. The number of integral points (integral point means both the coordinates should be integer) exactly in the interior of the triangle with vertices (0, 0), (0, 21)and (21, 0) is

1. 105
2. 133
3. 190
4. 233

Q.

Solve each of the following given systems of equations graphically and find the vertices and area of the triangle formed by these lines and the y-axis:

$2x-3y+6=0,2x+3y-18=0.$

Q. What is the area of the triangle formed by the points O (0, 0), A (6, 0) and B (0, 4)?

Q.

Draw the graph of $y=x^2-x-8$ and hence, solve the equation $x^2-2x-15=0$

Q.

If A (4, –6), B (3, –2) and C (5, 2) are the vertices of triangle ABC, then verify the fact that a median of a triangle ABC divides it into two triangle of equal areas