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The length of altitude through A of the triangle ABC where A = (-3, 0), B = (4, -1), C = (5, 2) is:

A) \( \Large \frac{2}{\sqrt{10}} \)
B) \( \Large \frac{4}{\sqrt{10}} \)
C) \( \Large \frac{11}{\sqrt{10}} \)
D) \( \Large \frac{22}{\sqrt{10}} \)

Correct Answer:




D) \( \Large \frac{22}{\sqrt{10}} \)

Description for Correct answer:

In \( \Large \triangle ABC \) the vertices are \( \Large A \left(-3,\ 0\right),\ B \left(4,\ -1\right)\ and\ C \left(5,\ 2\right) \)

Distance between B and C

\( \Large BC=\sqrt{ \left(5-4\right)^{2}+ \left(2+1\right)^{2} } = \sqrt{1+9} = \sqrt{10} \)

Area of \( \Large \triangle ABC \)

= \( \Large \frac{1}{2} \left[ x_{1} \left( y_{2} - y_{3} \right) + x_{2} \left( y_{3} - y_{1} \right) + x_{3} \left( y_{1} - y_{2} \right) \right] \)

= \( \Large \frac{1}{2}\left[ -3 \left(-1-2\right)+4 \left(2-0\right)+5 \left(0+1\right) \right] \)

= \( \Large \frac{1}{2}\left[ 9+8+5 \right]=11 \)

As we know area of \( \Large \triangle ABC = \frac{1}{2} \times BC \times AL \)

=> \( \Large 11 = \frac{1}{2} \times \sqrt{10} \times AL \)

=> \( \Large AL = \frac{2 \times 11}{\sqrt{10}}=\frac{22}{\sqrt{10}} \)

Part of solved Rectangular and Cartesian products questions and answers : >> Elementary Mathematics >> Rectangular and Cartesian products

Comments

Similar Questions

1). If orthocentre and circumcentre of triangle are respectively (1, 1) and (3, 2) then the co-ordinates of its centroid are:

A). \( \Large \left(\frac{7}{3},\ \frac{5}{3}\right) \)

B). \( \Large \left(\frac{5}{3},\ \frac{7}{3}\right) \)

C). (7, 5)

D). none of these

2). If
\( \begin{vmatrix}
x_{1} & y_{1} & 1 \\
x_{2} & y_{2} & 1 \\
x_{3} & y_{3} & 1
\end{vmatrix} = 0 \) , then the points \( \Large \left(x_{1},\ y_{1}\right),\ \left(x_{2},\ y_{2}\right)\ and\ \left(x_{3},\ y_{3}\right) \) are:

A). Vertices of an equilateral triangle

B). Vertices of a right angled triangle

C). Vertices of an isosceles triangle

D). none of these

3). If two vertices of an equilateral triangle are \( \Large \left(0,\ 0\right)\ and\ \left(3,\ 3\sqrt{3}\right) \) then the third vertex is:

A). \( \Large \left(3,\ -3\right) \)

B). \( \Large \left(-3,\ 3\right) \)

C). \( \Large \left(-3,\ 3\sqrt{3}\right) \)

D). none of these

4). If origin is shifted to \( \Large \left(7,\ -4\right) \) then point \( \Large \left(4,\ 5\right) \) shifted to

A). \( \Large \left(-3,\ 9\right) \)

B). \( \Large \left(3,\ 9\right) \)

C). \( \Large \left(11,\ 1\right) \)

D). none of these

5). The feet of the perpendicular drawn from P to the sides of a triangle ABC are collinear, then P is:

A). circumcentre of triangle ABC

B). lies on the circumcircle of triangle ABC

C). excentre of triangle ABC

D). none of these

6). The points \( \Large \left(k,\ 2-2k\right) \), \( \Large \left(-k+1,\ 2k\right) \), \( \Large \left(-4-k,\ 6-2k\right) \) are collinear then k is equal to:

A). 2, 3

B). 1, 0

C). \( \Large \frac{1}{2},\ 1 \)

D). 1, 2

7). Let AB is divided internally and externally at P and Q in the same ratio. Then AP, AB, AQ are in

A). AP

B). GP

C). HP

D). none of these

8). If O be the origin and if \( \Large P_{1} \left(x_{1},\ y_{1}\right)\ and\ P_{2} \left(x_{2},\ y_{2}\right) \) be two points, then \( \Large \left(OP_{1} \parallel \ OP_{2} \right) \cos \left( \angle P_{1}\ OP_{2}\right) \) is equal to:

A). \( \Large x_{1}y_{2}+x_{2}y_{1} \)

B). \( \Large \left(x^{2}_{1}+x^{2}_{2}+y^{2}_{2}\right) \)

C). \( \Large \left(x_{1}-x_{2}\right)^{2}+ \left(y_{1}-y_{2}\right)^{2} \)

D). \( \Large x_{1}x_{2}+y_{1}y_{2} \)

9). The median BE and AD of triangle with vertices \( \Large A \left(0,\ b\right), B \left(0,\ 0\right)\ and\ C \left(a,\ 0\right) \) are perpendicular to each other if;

A). \( \Large a=\frac{b}{2} \)

B). \( \Large b=\frac{a}{2} \)

C). ab = 1

D). \( \Large a\ =\ \pm \sqrt{2}b \)

10). ABC is a triangle with vertices A = (-1, 4), B = (6, -2) and C = (-2, 4). D, E and F are points which divide each AB, BC and CA respectively in the ratio 3 : 1 internally. Then centroid of the triangle DEF is:

A). \( \Large \left(3,\ 6\right) \)

B). \( \Large \left(1,\ 2\right) \)

C). \( \Large \left(4,\ 8\right) \)

D). \( \Large \left(-3,\ 6\right) \)