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
Let n be the number. The difference of the number and six is n-6. Then three times the difference of the number and six is 3·(n-6). This number 3·(n-6) increased by four times the number n, then
3·(n-6)=4·n.
Solve it:
3·n-3·6=4n,
3n-4n=18,
-n=18,
n=-18.
Answer: n=-18 and expression is 3·(n-6)=4·n.
You know the long leg<span> (the </span>side<span> across from the </span>60<span>-degree angle). Divide this </span>side<span> by the square root of 3 to </span>find<span> the short </span>side<span>. Double that figure to </span>find<span> the hypotenuse. </span>Finding<span> the other </span>sides<span> of a </span>30-60-90 triangle<span> when you know the hypotenuse.</span>
When slicing a cube it is impossible to create a circular cross section and an octagonal cross section. The reason for this is, a cube only has 6 sides and so no matter how many ways you slice it you can't get a shape with more sides than that.
In this problem, we use the method of ratio and proportion. This technique works by using the concept that there is a fixed ratio of compositions within a mixture. Speaking generally, there is a fixed ratio of the parts of the whole. So, all we have to do is equate both ratios.
5 L natural oil/ 6 L synthetic oil = x L natural oil/498 L synthetic oil
x = 415 L natural oil
So, if petrol oil contains 498 L synthetic oil, it has a corresponding amount of 415 natural oil. Adding these two would be the total volume of the petrol oil. That would be 415+498 = 913 L petrol oil.
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