PLZ help going crazy here with this question.?

Question

an equilateral triangle with sides 10cm is divided into two regions by a line segment parallel to one of the sides.

a) if the regions have equal areas, determine the length of the line segment.

b) create two other problems that are suggested by this one. then solve each problem.

okay so i figured the height of the big triangle to be 8.7cm. and the area is 43cm squared. This means the area of the small triangle should be 21.7cm squared.

im stuck here... any help is appreciated. thanx alot . for the second part i have no idea so if u can help thanx :)

Answer 1

An equilateral triangle with side length s has area A=[(sqrt3)/4]s^2. In other words, the area is proportional to the square of the side length. (A = k*s^2, where k is the constant (sqrt3)/4, approx. 0.433)

So, the large triangle has area k*10^2 = 100k. Therefore the smaller triangle must have half this area, or 50k. This means that for the smaller triangle, s^2=50, so the side length is sqrt50, or 5sqrt2. This value is approx. 7.071067812 cm.

Sorry - you're on your own for part b! Maybe you could design a similar problem using isosceles triangles, or even a square that's cut into 2 rectangles? Have fun!

Answer 2

Ok, this was kind of tough, but:

Part A: total triangle area = 43.3, so each of the new regions must have an area of 21.65. When the triangle is split, it has two parts: a trapezoid and a smaller triangle. the height of the trapezoid = h1, the height of the triangle = h2. both share the same base (b). The formula for a trapezoid = (1/2(b1+b2))*h. In this case, it is .5(10 + b )h1.
The formula for this triangle is .5(b*h2).
We also know that h1 = 8.66-h2 and that equation 1 (trapezoid) = equation 2 (triangle). So we can substitute (8.66-h2) into the trap. equation for h1 and set the equations equal to each other. Solve for h2 and you will get:

h2 = (86.6 - 8.66b) / (10 + 1.5b)

Now plug this into the equation for the small triangle:
21.65 = .5h2*b

plug in h2, solve for b (you will have to use the quadratic formula) and you should get: b= 6.06 or b= 2.06. But we know by sketching a quick picture that b must be 6.06. This b is the length of the line segment.

Hope this helps!! Part b is just up to you.

SCRATCH THAT. I DID THE MATH WRONG, BUT THE SEGMENT TURNS OUT TO BE 5.68.

Answer 3

Area = 1/2side^2*sqrt(3 )/2 as height = side *sqrt(3)2
so accepting your numbers
21.7 = side^2* sqrt3/4
and side= sqrt( 4*21.7/sqrt3)=7.08cm