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Multiple choice maths constructions mid-point formula midpoints division of a line segment

Find the area of the triangle formed by joining the mid points of the sides of the triangle whose vertices are $(0.-1), (2, 1) and (0, 3)$

  1. $4$
  2. $8$
  3. $1$
  4. $2$
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

$\\Area\>of\>triangle\>=4\times\>of\>triangle\>formed\>using\>mid-point\>\\=4\times(\frac{1}{2})[x-1(y _2-y _3)+x _2(y _3+y _1)+x _3(y _1-y _2)]\\=2[0+2(3-1)+0]=8sq\>unit$

Multiple choice maths constructions mid-point formula midpoints division of a line segment

What is the y intercept of the line that is parallel to $y=3x,$ and which bisects the area of rectangle with corners at $(0,0), (4,0) ,(4,2) $ and $(0,2)$? 

  1. $ -7$
  2. $-6$
  3. $ -5$
  4. $ -4$
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

Rectangle's midpoints are $ = \left( {\frac{{0 + 4}}{2},\frac{{0 + 2}}{2}} \right) = \left( {\frac{4}{2},\frac{2}{2}} \right) = \left( {2,1} \right)$

Slope line y$=$ 3x will be :
${m _1} = 3$
Parallel line will be
${m _2} = 3$
Equation of line passing through (2 , 1)
$y - {y _1} = m(x - {x _1})$
$(y - 1) = 3(x - 2)$
$y - 1 = 3x - 6$
$y = 3x - 5$
Hence on comparing 
Y- intercept $=$ - 5

Multiple choice maths constructions mid-point formula midpoints division of a line segment

The mid-point of the line $(a, 2)$ and $(3, 6)$ is $(2, b)$. Find the numerical values of $a$ and $b$.

  1. $a=1$, $b=6$
  2. $a=2$, $b=4$
  3. $a=1$, $b=4$
  4. $a=2$, $b=6$
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

Mid-point of $(a,2)$ and $(3,6)$ is $(2,b)$

=>$(2,b)=\left( \cfrac { a+3 }{ 2 } ,\cfrac { 2+6 }{ 2 }  \right) \ =>a=4-3,b=4\ =>a=1,b=4$

Multiple choice maths constructions mid-point formula midpoints division of a line segment

If $(3, -4)$ and $(-6, 5)$ are the extremities of a diagonal of a parallelogram and $(2, 1)$ is its third vertex, then its fourth vertex is?

  1. $(-1, 0)$
  2. $(-1, 1)$
  3. $(0, -1)$
  4. $(-5, 0)$
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation

In a parallelogram, the diagonals bisect each other. Let the vertices be A(3, -4), B(x, y), C(-6, 5), and D(2, 1). The midpoint of diagonal AC is ((3-6)/2, (-4+5)/2) = (-1.5, 0.5). The midpoint of diagonal BD is ((x+2)/2, (y+1)/2). Setting these equal gives x+2 = -3 (x=-5) and y+1 = 1 (y=0).

Multiple choice maths constructions mid-point formula midpoints division of a line segment

If $(6, -3)$ is the one extremity of diameter to the circle $x^{2}+y^{2}-3x+8y-4=0$ then its other extremity is-

  1. $(3/2, -4)$
  2. $(-3, -5)$
  3. $(3, -5)$
  4. $(3, 5)$
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

The center of the circle x^2 + y^2 - 3x + 8y - 4 = 0 is (3/2, -4). Since the center is the midpoint of the diameter, if one end is (6, -3), let the other be (x, y). Then (6+x)/2 = 3/2 implies x = -3, and (-3+y)/2 = -4 implies y = -5.

Multiple choice maths constructions mid-point formula midpoints division of a line segment

The length of the median from the vertex A of a triangle whose vertices are $A (-1, 3),$ B $(1, -1)$ and C$(5,1)$ is 

  1. $5$
  2. $4$
  3. $1$
  4. $3$
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation
Length of the median from the vertex $A$ of a triangle $\triangle{ABC}$
Let $AD$ be the median.

$\Rightarrow\,D$ is the midpoint of $BC$

Using midpoint formula,$D=\left(\dfrac{1+5}{2},\,\dfrac{-1+1}{2}\right)=\left(3,\,0\right)$

Length of median $=AD=\sqrt{{\left(-1-3\right)}^2{}+{\left(0-3\right)}^{2}}=\sqrt{16+9}=\sqrt{25}=5$units.
Multiple choice maths constructions mid-point formula midpoints division of a line segment

The locus of mid points of chords to the circle $x^{2}+y^{2}-8x+6y+20=0$ which are parallel to the line $3x+4y+5=0$ 

  1. $3x+4y-25=0$
  2. $4x+3y+5=0$
  3. $4x-3y-25=0$
  4. $4x-3y+25=0$
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

The locus of the midpoints of chords of a circle parallel to a given line is a diameter of the circle perpendicular to the given line or parallel to the normal direction. Given the circle equation and the parallel line 3x + 4y + 5 = 0, calculating the center and finding the line passing through it with the perpendicular slope yields 4x - 3y - 25 = 0.

Multiple choice maths constructions mid-point formula midpoints division of a line segment

If $(2, 3), (-4, 5), (1, -2)$ are the midpoints of the sides $\vec{BC}, \vec{CA}, \vec{AB}$ of $\triangle ABC$, then the equation of $\vec{AB}$ is 

  1. $3x-y-5=0$
  2. $x+3y+5=0$
  3. $x+3y-11=0$
  4. $3x-y+17=0$
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

Let vertices be A(x1, y1), B(x2, y2), C(x3, y3). Using the given midpoints, set up midpoint equations to find the coordinates of vertices A and B. Once vertices A and B are determined, find the equation of the line passing through them, which results in x + 3y + 5 = 0.

Multiple choice maths constructions mid-point formula midpoints division of a line segment

The point on $X-axis$ equidistant from $(2,3)$and $(1,5)$ is

  1. $\left( \dfrac { -13 }{ 2 } ,0 \right) $
  2. $\left( \dfrac { 13 }{ 2 } ,0 \right) $
  3. $(13,0)$
  4. $none\ of\ these$
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

Let the point be (x, 0). Equidistance means (x-2)^2 + (0-3)^2 = (x-1)^2 + (0-5)^2. Expanding: x^2 - 4x + 4 + 9 = x^2 - 2x + 1 + 25. Simplifying: -4x + 13 = -2x + 26, so -2x = 13, x = -13/2. The point is (-13/2, 0).