Let x=ab=ac, and y=bc, and z=ad.
Since the perimeter of the triangle abc is 36, you have:
Perimeter of abc = 36
ab + ac + bc = 36
x + x + y = 36
(eq. 1) 2x + y = 36
The triangle is isosceles (it has two sides with equal length: ab and ac). The line perpendicular to the third side (bc) from the opposite vertex (a), divides that third side into two equal halves: the point d is the middle point of bc. This is a property of isosceles triangles, which is easily shown by similarity.
Hence, we have that bd = dc = bc/2 = y/2 (remember we called bc = y).
The perimeter of the triangle abd is 30:
Permiter of abd = 30
ab + bd + ad = 30
x + y/2 + z =30
(eq. 2) 2x + y + 2z = 60
So, we have two equations on x, y and z:
(eq.1) 2x + y = 36
(eq.2) 2x + y + 2z = 60
Substitute 2x + y by 36 from (eq.1) in (eq.2):
(eq.2') 36 + 2z = 60
And solve for z:
36 + 2z = 60 => 2z = 60 - 36 => 2z = 24 => z = 12
The measure of ad is 12.
If you prefer a less algebraic reasoning:
- The perimeter of abd is half the perimeter of abc plus the length of ad (since you have "cut" the triangle abc in two halves to obtain the triangle abd).
- Then, ad is the difference between the perimeter of abd and half the perimeter of abc:
ad = 30 - (36/2) = 30 - 18 = 12
Answer: 35
Step-by-step explanation:
Triangle BDC is an equilateral triangle
Triangle ABD is an isosceles triangle
so the length of BD is 8
because BDC is equilateral all sides are 8
then you add your numbers up
8+8+8+11
=35
F(x) = -4x + 1
g(x) = 3x + 1
A. (f + g)(x) = (-4x + 1) + (3x + 1)
(f + g)(x) = (-4x + 3x) + (1 + 1)
(f + g)(x) = x + 2
Domain: (-∞, ∞) {x|x ∈ R}
B. (f - g)(x) = (-4x + 1) - (3x + 1)
(f - g)(x) = (-4x - 3x) + (1 - 1)
(f - g)(x) = -7x
Domain: (-∞, ∞) {x|x ∈ R}
C. (f · g)(x) = (-4x + 1)(3x + 1)
(f · g)(x) = -4x(3x + 1) + 1(3x + 1)
(f · g)(x) = -4x(3x) - 4x(1) + 1(3x) + 1(1)
(f · g)(x) = -12x² - 4x + 3x + 1
(f · g)(x) = -12x² - x + 1
Domain: (-∞, ∞) {x|x ∈ R}
D. (f ÷ g)(x) = (-4x + 1) ÷ (3x + 1)
Domain: (-∞, ⁻¹/₃) U (⁻¹/₃, ∞) {x|x ≠ ⁻¹/₃}
