Answer:
(a) square: L; triangle: 0.
(b) square: L·(-16+12√3)/11; triangle: L·(27-12√3)/11
Step-by-step explanation:
<u>Strategy</u>: First we will write each area in terms of its perimeter. Then we will find the total area in terms of the amount devoted to the square. Differentiating will give a way to find the minimum total area.
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In terms of its perimeter p, the area of a square is ...
A_square = p^2/16
In terms of its perimeter p, the area of an equilateral triangle is ...
A_triangle = p^2/(12√3)
Then the total area of the two figures whose total perimeter is L with "x" devoted to the square is ...
A_total = x^2/16 + (L-x)^2/(12√3)
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(a) We know when polygons are regular, the one with the most area for the least perimeter is the one with the most sides. Hence, the total area is maximized when all of the wire is devoted to the square.
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(b) The derivative of A_total with respect to x is ...
dA/dx = x/8 -(L-x)/(6√3)
This will be zero when ...
x/8 = (L-x)/(6√3)
x(6√3) = 8L -8x
x(8 +6√3) = 8L
x = L·8/(6√3 +8) = 8L(6√3 -8)/(64-108)
x = L·(12√3 -16)/11
The total area is minimized when L·(12√3 -16)/11 is devoted to the square, and the balance is devoted to the triangle.
Answer:
∠1 and ∠2
angles on a straight line add up to 180
Answer:
Step-by-step explanation:
1) In order to solve a system of equations with the elimination method, you need to add or subtract the two equations and "eliminate" variables by canceling out terms. This leaves you with an equation with only one variable to solve. Then, once you solve that equation, you can substitute the answer you got into one of the system's equations to find either the x or y value.
This problem already has one 3x in each equation, so let's focus on canceling that term out, eliminating the x variable. To set up this situation, we need one 3x to be negative, so let's multiply the terms in 3x-6y = 4 by -1, which gets us to -3x + 6y = -4. Now let's "cancel out" those x variables by adding them together. (This work is shown in the first picture.)
Remember to add the other terms. This leaves us with the equation 13y = 13. By isolating y, we can find its value, which leaves us with y = 1.
2) Next, let's substitute 1 for y in one of the two equations. In this case, I used 3x + 7y = 17. Then, isolate x to find its value:
Therefore, the answer would be 
.
Answer:
19/24
Step-by-step explanation:
find the common denominator
How many facts does it take to make triangles congruent? Only 3 if they are the right three and the parts are located in the right place.
SAS where 2 sides make up one of the three angles of a triangle. The angle must between the 2 sides.
ASA where the S (side) is common to both the two given angles.
SSS where all three sides of one triangle are the same as all three sides of a second triangle. This one is my favorite. It has no exceptions.
In one very special case, you need only 2 facts, but that case is very special and it really is one of the cases above.
If you are working with a right angle triangle, you can get away with being given the hypotenuse and one of the sides. So you only need 2 facts. It is called the HL theorem. But that is a special case of SSS. The third side can be found from a^2 + b^2 = c^2.
You can also use the two sides making up the right angle but that is a special case of SAS.
Answer
There 6 parts to every triangle: 3 sides and 3 angles. If you show congruency, using any of the 3 facts above, you can conclude that the other 3 parts of the triangle are congruent as well as the three that you have.
Geometry is built on that wonderfully simple premise and it is your introduction to what makes a proof. So it's important that you understand how proving parts of congruent triangles work.
