Question

ABC is an equilateral tri-
angle. Point D is on AC and
point E is on BC, such that
AD = 2CD and CE = EB.
D
If we draw perpendiculars ch
B
from D and E to other two
E
sides and find the sum of the length of two perpen-
diculars for each set, that is, for D and E individually
and denote them as per (D) and per (E) respectively,
then which of the following option will be correct.
(a) Per (D) > per (E)
(b) Per (D) < per (E)
(c) Per (D) = per (E)
(d) Cannot be determined​

Answer 1

Answer:

is this Aryan........

Answer 2

Answer:

Let note alpha the measure of the angle of ΔABC and beta the length of the side. We will note as well the length of the projections from D and E, with x, y, and z, w, respectively, such as Per (D) = x+y and Per (E) = z + w.

(see the image)

Using trigonometry we compute each of those, function of alpha and beta only using the Formula: sin (angle) = opposite / hypotenuse,

Step-by-step explanation:

thus:

sin(alpha) = x / (beta* 1/3) => x = (beta * 1/3) * sin(alpha)

sin(alpha) = y / (beta * 2/3) = > y = (beta * 2/3) * sin(alpha), and:

sin(alpha) = z / (beta * 1/2) => z = (beta * 1/2) * sin(alpha)

sin(alpha) = w / (beta * 1/2) => w = (beta * 1/2) * sin(alpha)

Then:

x + y = beta * sin(alpha), and w + z = beta * sin(alpha)

So: Per (D) = Per (E) Q.E.D.

Another way to prove this, by not involving trigonometry, is to split ΔABC into two triangles ΔBDC, ΔBDA where x and y are altitudes, and simply compute the area of those triangles [ΔArea = (Base * Altitude) / 2], which sums-up to the area of ΔABC which is constant. Do the same for ΔBEA, ΔCEA, where w and z are the altitudes, this time.

Since the sum of the areas is constant AND the side is constant => the sum of the altitudes is constant. Q.E.D.

As a generalization, we can say that for an equilateral triangle, the sum of the length of the perpendiculars drawn on two sides from the third side will be constant. This is a particular case of Viviani's Theorem: The sum of the distances from any interior point to the sides of an equilateral triangle equals the length of the triangle's altitude.

Figure drawn using GeoGebra geogebra dot org

PS sorry for any of my English mistakes