An equalateral triangle has the length of it's sides given as (y+3)cm, (2y-x)cm and (4x +3)cm.Find (a) the value of x and y (b)perimeter of the triangle (c) altitude of the triangle to the nearest whole number

all 3 sides are equal , so form equations using any two pairs.

y+3 = 2y-x ---> y = 3+x

y+3 = 4x+3 --> y = 4x

so 4x = 3+x
3x=3
x = 1

then y = 4(1) = 4

x=1 , y=4

each side is 7 , perimeter is 21
for the height, use Pythagoras
3.5^2 + h^2 = 7^2
h = √36.75 = appr. 6.06

or

sin60° = h/7
h = 7sin60 = 6.06

all 3 sides are equal , so form equations using any two pairs.

y+3 = 2y-x ---> y = 3+x

y+3 = 4x+3 --> y = 4x

so 4x = 3+x
3x=3
x = 1

then y = 4(1) = 4

x=1 , y=4

each side is 7 , perimeter is 21
for the height, use Pythagoras
3.5^2 + h^2 = 7^2
h = √36.75 = appr. 6.06

or

sin60° = h/7
h = 7sin60 = 6.06

To solve this problem, we need to use the properties of an equilateral triangle, where all sides are equal in length. We can set the expressions for the sides equal to each other and solve for the variables x and y.

(a) To find the values of x and y, we can set up equations based on the given side lengths:

(y + 3) = (2y - x) = (4x + 3)

First, let's compare the first two expressions:

y + 3 = 2y - x

Rearranging the terms, we get:

x = 2y - (y + 3)
x = 2y - y - 3
x = y - 3

Now, we can substitute this value of x into the third expression:

4x + 3 = 4(y - 3) + 3
4x + 3 = 4y - 12 + 3
4x + 3 = 4y - 9

Comparing this equation to the first equation, we have:

4y - 9 = (4x + 3)

Setting the two expressions equal to each other, we get:

4y - 9 = 2y - x
4y - 2y = -x + 9
2y = -x + 9
y = (-x + 9) / 2

Now, we can substitute this value of y back into the first equation:

x = y - 3
x = [(-x + 9) / 2] - 3

Simplifying further:

x = -x/2 + 9/2 - 3
x = -x/2 - 3/2 + 9/2
x + x/2 = 9/2 - 3/2
(3/2)x = 6/2
(3/2)x = 3
x = 2

Finally, we can substitute this value of x back into the equation for y:

y = (-x + 9) / 2
y = (-(2) + 9) / 2
y = 7/2

So, the values of x and y are x = 2 and y = 7/2.

(b) Now that we have the values of x and y, we can find the perimeter of the equilateral triangle. The perimeter is the sum of all three sides:

Perimeter = (y + 3) + (2y - x) + (4x + 3)
Perimeter = (7/2 + 3) + (2 * 7/2 - 2) + (4 * 2 + 3)
Perimeter = (7/2 + 6) + (7 - 2) + (8 + 3)
Perimeter = 13/2 + 5 + 11
Perimeter = 37/2

Therefore, the perimeter of the equilateral triangle is 37/2 cm.

(c) To find the altitude of the triangle, we can use the formula for the area of an equilateral triangle:

Area = (side^2 * √3) / 4

Given that side = (y + 3) cm, we can substitute the value of y:

Area = [(7/2 + 3)^2 * √3] / 4
Area = [(29/2)^2 * √3] / 4
Area = (841/4 * √3) / 4
Area = (841√3) / 16

Now, we can find the altitude using the formula:

Area = (base * altitude) / 2

Substituting the values:

(841√3) / 16 = [(y + 3) * altitude] / 2

Simplifying further:

altitude = (841√3 * 2) / [16 * (y + 3)]
altitude = (841√3) / (8y + 24)

Now, let's substitute y = 7/2:

altitude = (841√3) / (8(7/2) + 24)
altitude = (841√3) / (28/2 + 24)
altitude = (841√3) / (28 + 24)
altitude = (841√3) / 52

Rounding this value to the nearest whole number, we get:

altitude ≈ 25

Therefore, to the nearest whole number, the altitude of the equilateral triangle is 25 cm.