Question

$AB$ and $CD$ are $2$ line segments ; where $A(2,3,0),B (6, 9, 0), C(-6, -9, 0). P $ and $Q$ are midpoint of $AB$ and $CD$, respectively and $L$ is the midpoint of $PQ$. Find the distance of $L$ from the plane $3x + 4z + 25 = 0$

• A. 25
• B. 15
• C. 5
• D. 40

Answer 1

The Correct Option is C

Let co-ordinate of $D$ is $(x, y, z)$
Using mid-point formula
$Q=\left(\frac{x-6}{2}, \frac{y-9}{2} , \frac{z+0}{2}\right) $
Also, $P=\left(\frac{2+6}{2}, \frac{3+9}{2}, \frac{0+0}{2}\right) =\left(4,6,0\right)$
Since, $AC||PQ$
$ \therefore$ D.r's of line $AC $ = D.r's of line $PQ$
$\Rightarrow \:\: \left(-8 ,12,0\right)=\left(\frac{x-14}{2}, \frac{y-21}{2} , \frac{z}{2}\right) $
$\Rightarrow\:\: x = -2, y = -3, z = 0$
$\Rightarrow\:\:\: D (-2, -3,0) \:\:\: \Rightarrow \:\: Q(-4, - 6,0)$
If $L$ is midpoint of $PQ$ then
$L\left(\frac{4-4}{2}, \frac{6-6}{2}, 0\right)=\left(0,0,0\right)$
$ \therefore$ $ \perp$ distance of $L(0,0,0)$ from the plane $3x + 4z + 25 = 0$ is
$=\left|\frac{3\left(0\right)+4\left(0\right)+25}{\sqrt{\left(3\right)^{2}+\left(4\right)^{2}}}\right|= \left|\frac{25}{\sqrt{25}}\right| =\sqrt{25} =5$

Answer 2

Ans. The word "midpoint" has been used to refer to the exact "center" of anything since the early 14th century. The center point is where the two ends meet. It may also be referred to as the line segment's centroid and its ends. 

The midpoint theorem states that "The line segment in a triangle joining the midpoint of two sides of the triangle is said to be parallel to its third side and is also half of the length of the third side."

The midpoint theorem came to be invented by Rene Descartes, the most popular mathematician of all time. It was a basic geometrical idea at first in Rene’s mind, which represented an ordered pair of numbers. This was later made into a method to help combine arithmetic and geometry. Apart from the cartesian plane, the midpoint has great uses in geometrical figures like triangles, circles, ellipses, hyperbolas, quadrilaterals, and general polygons.

Let’s suppose that a line connects two points (2,6) and (4,2), then the coordinates of the midpoint of the line joining these two points are, [(2+4)/2, (6+2)/2] which gives us (3,4).