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
100 miles / 4 hours
(divide each side by 4)
25 miles / hour
44/11= 4
4 minutes of the movie are downloaded each minute.
1.5 hours= 90 min
90/4= 22.5 minutes
For any polynomial, first check to see if it factors. Then you can set each factor equal to zero.
The three zeros are -1,1,3
Next plug in values in each interval between zeros to test whether its greater than 0.
Try x values: -2,0,2,4
x = -2 yields a negative result, the interval x< -1 is Not a solution.
x = 0 yields a positive result, the interval -1
x = 2 yields a negative result, the interval 1
x = 4 yields a positive result, the interval x>3 IS a solution
