Advertisements
Advertisements
Question
The area of a triangle with vertices A, B, C is given by ______.
Options
`|vec("AB") xx vec("AC")|`
`1/2|vec("AB") xx vec("AC")|`
`1/4|vec("AC") xx vec("AB")|`
`1/8|vec("AC") xx vec("AB")|`
Solution
The area of a triangle with vertices A, B, C is given by `underline(bb(1/2|vec(AB) xx vec(AC)|))`.
Explanation:
The area of the parallelogram with adjacent sides AB and AC = `|vec("AB") xx vec("AC")|`. Hence, the area of the triangle with vertices A, B, C = `1/2|vec("AB") xx vec("AC")|`
APPEARS IN
RELATED QUESTIONS
The perimeter of a right triangle is 60 cm. Its hypotenuse is 25 cm. Find the area of the triangle.
If the points A(x, 2), B(−3, −4) and C(7, − 5) are collinear, then the value of x is:
(A) −63
(B) 63
(C) 60
(D) −60
If the points A(−1, −4), B(b, c) and C(5, −1) are collinear and 2b + c = 4, find the values of b and c.
median of a triangle divides it into two triangles of equal areas. Verify this result for ΔABC whose vertices are A (4, - 6), B (3, - 2) and C (5, 2).
Find the centre of a circle passing through the points (6, − 6), (3, − 7) and (3, 3).
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)
ABCD is a rectangle formed by the points A(-1, -1), B(-1, 4), C(5, 4) and D(5, -1). P, Q, R and S are the midpoints of AB, BC, CD and DA respectively. Is the quadrilateral PQRS a square? a rectangle? or a rhombus? Justify your answer.
Find the area of a triangle with vertices at the point given in the following:
(1, 0), (6, 0), (4, 3)
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 (3, 1) and (9, 3) using determinant.
Find the area of the following triangle:
The vertices of ΔABC are (−2, 1), (5, 4) and (2, −3) respectively. Find the area of the triangle and the length of the altitude through A.
Find the angle subtended at the origin by the line segment whose end points are (0, 100) and (10, 0).
Prove that the lines joining the middle points of the opposite sides of a quadrilateral and the join of the middle points of its diagonals meet in a point and bisect one another
If G(-2, 1) is the centroid of a ΔABC and two of its vertices are A(1, -6) and B(-5, 2) , find the third vertex of the triangle.
Prove that the points A (a,0), B( 0,b) and C (1,1) are collinear, if `( 1/a+1/b) =1`.
Find the value of p for which the points (−5, 1), (1, p) and (4, −2) are collinear.
Using integration, find the area of triangle ABC, whose vertices are A(2, 5), B(4, 7) and C(6, 2).
Find BC, if the area of the triangle ABC is 36 cm2 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.
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.
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 ∆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
Find the area of the triangle whose vertices are (-2, 6), (3, -6), and (1, 5).
The area of the triangle ABC with the vertices A(-5, 7), B(-4, -5) and C(4, 5) is ______.
The area of the triangle whose vertices are A(1, 2), B(-2, 3) and C(-3, -4) is ______.
Points A(–6, 10), B(–4, 6) and C(3, –8) are collinear such that AB = `2/9` AC.
The area of a trapezium is 475 cm2 and the height is 19 cm. Find the lengths of its two parallel sides if one side is 4 cm greater than the other.
