Tag: introduction to 3d geometry

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Multiple choice maths introduction to three dimensional geometry coordinate axes and coordinate planes in 3d space coordinates in 3d introduction to 3d geometry

If G is the centroid of $\triangle ABC$ and BC = 3, CA = 4, AB = 5 then BG =

  1. $\dfrac { \sqrt { 73 } }{ 3 } $

  2. $\dfrac { \sqrt { 13 } }{ 3 } $

  3. $\dfrac { \sqrt { 52 } }{ 3 } $

  4. $\dfrac { \sqrt { 26 } }{ 3 } $
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

The triangle with sides 3, 4, 5 is a right triangle. Using Apollonius theorem or coordinate geometry, the length of the median to side AB (c=5) is m_c = 1/2 * sqrt(2a^2 + 2b^2 - c^2) = 1/2 * sqrt(2*16 + 2*9 - 25) = 1/2 * sqrt(7) = sqrt(7)/2. The centroid G divides the median in a 2:1 ratio, so BG = 2/3 * m_c = 2/3 * sqrt(7)/2 = sqrt(7)/3. However, checking the options, sqrt(52)/3 is 2*sqrt(13)/3. Let's re-verify: median to AB is 1/2 * sqrt(2*16 + 2*9 - 25) = sqrt(7)/2. The distance BG is 2/3 of the median. None match perfectly. Re-evaluating: maybe median to BC? m_a = 1/2 * sqrt(2*16 + 2*25 - 9) = 1/2 * sqrt(32+50-9) = sqrt(73)/2. BG = 2/3 * sqrt(73)/2 = sqrt(73)/3. Option A is sqrt(73)/3. Wait, the question asks for BG, which is the segment from vertex B to centroid G. This is 2/3 of the median from B to AC. Median m_b = 1/2 * sqrt(2*a^2 + 2*c^2 - b^2) = 1/2 * sqrt(2*9 + 2*25 - 16) = 1/2 * sqrt(18+50-16) = 1/2 * sqrt(52) = sqrt(52)/2. BG = 2/3 * sqrt(52)/2 = sqrt(52)/3.

Multiple choice maths introduction to three dimensional geometry coordinate axes and coordinate planes in 3d space coordinates in 3d introduction to 3d geometry

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

  1. $( - 1,0 )$

  2. $( - 1,1 )$

  3. $( 0,-1 )$

  4. $( 7,8 )$
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation

In a parallelogram, the diagonals bisect each other. The midpoint of the diagonal with endpoints (3,4) and (6,5) is ((3+6)/2, (4+5)/2) = (4.5, 4.5). Let the fourth vertex be (x,y). The midpoint of the other diagonal (2,1) and (x,y) must also be (4.5, 4.5). So (2+x)/2 = 4.5 => x = 7, and (1+y)/2 = 4.5 => y = 8. The fourth vertex is (7,8).

Multiple choice maths introduction to three dimensional geometry coordinate axes and coordinate planes in 3d space coordinates in 3d introduction to 3d geometry

The foot of the perpendicular from the point $A(7, 14, 5)$ to the plane $2x+4y-z=2$ is?

  1. $(3, 1, 8)$

  2. $(1, 2, 8)$

  3. $(3, -3, 5)$

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

Let N be the foot of the perpendicular drawn from the point $A(7, 14, 5)$ and perpendicular to the plane $2x+4y-z=2$

Then, the equation of the line PN is $\dfrac{x-7}{2}=\dfrac{y-14}{4}=\dfrac{z-5}{-1}=\lambda$ (say)
Let the coordinates of N be $N(2\lambda +7, 4\lambda +14, -\lambda +5)$
Since N lies on the plane $2x+4y-z=2$, so
$2(2\lambda +7)+4(4\lambda +14)-(-\lambda +5)=2$
$\Rightarrow 21\lambda =-63$
$\Rightarrow \lambda =-3$
$\therefore$ required foot of the perpendicular is
$N(-6+7, -12+14, 3+5)$, i.e., $N(1, 2, 8)$.

Multiple choice maths introduction to three dimensional geometry coordinate axes and coordinate planes in 3d space coordinates in 3d introduction to 3d geometry

In geometry, we take a point, a line and a plane as undefined terms.

  1. True

  2. False

  3. Ambiguous

  4. Data Insufficient
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

 In Geometry, we define a point as a location and no size. A line is defined as something that extends infinitely in either direction but has no width and is one dimensional while a plane extends infinitely in two dimensions.

Multiple choice maths introduction to three dimensional geometry coordinate axes and coordinate planes in 3d space coordinates in 3d introduction to 3d geometry

Arrange the points: $\mathrm{A}(1,2-3), \mathrm{B}(-1,2,-3), \mathrm{C}(-1,-2-3)$ and $\mathrm{D}(1,-2, -3)$ in the increasing order of their octant numbers:

  1. $A,B,C,D$

  2. $B,C,D,A$

  3. $C,D,A,B$

  4. $D,C,B,A$
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation
 Octant $I$  $II$ $III$  $IV$  $V$  $VI$  $VII$ $VIII$
 Signs: $+,+,+$  $-,+,+$ $-,-,+$  $+,-,+$  $+,+,-$ $-,+,-$ $-,-,-$  $+,-,-$ 

Based on this, increasing order is

$ A,B ,C,D$

Multiple choice maths introduction to three dimensional geometry coordinate axes and coordinate planes in 3d space coordinates in 3d introduction to 3d geometry

Graph $x^2+y^2=4$ in 3D looks like

  1. Circle

  2. Cylinder

  3. Hemisphere

  4. Sphere
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

The given curve is $x^2+y^2=4$ 

So $x$ coordinate and y-coordinate are connected by $x^2+y^2=4$
which is locus of a circle with radius $2$
But z-coordinate can be anything, so in three dimension the circle $x^2+y^2=4$ will be 
stretched which will be a cylinder with radius same as the radius of the circle .

Multiple choice maths introduction to three dimensional geometry coordinate axes and coordinate planes in 3d space coordinates in 3d introduction to 3d geometry

An equation of sphere with centre at origin and radius $r$ can be represented as

  1. $x^2+y^2+z^2=r$

  2. $x^2+y^2+z^2=r^2$

  3. $x^2+y^2+z^2=2r^2$

  4. None of the above
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

Sphere is locus of a point in 3D whose distance from a fixed point(center) is constant (radius)

$\Rightarrow \sqrt{(x-0)^2+(y-0)^2+(z-0)^2}=|r|$
$\Rightarrow x^2+y^2+z^2=r^2$, square both sides 

Multiple choice maths introduction to three dimensional geometry coordinate axes and coordinate planes in 3d space coordinates in 3d introduction to 3d geometry

The equation of plane passing through $(-1,0,-1)$ parallel to $xz$ plane is

  1. $y=-2$

  2. $y=0$

  3. $-x-z=0$

  4. None of the above
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

Given that the plane is parallel to $xz$ plane and the plane passes through $(-1,0,-1)$

Since the plane is parallel to $xz$ plane , the $y$ coordinate should be constant
Given that it passes through point $(-1,0,-1)$ , therefore the plane lies on $xz$ plane
Therefore the equation of plane is $y=0$
The correct options are $B$