Notice the picture
the rhombus, or kite, is really just 4 triangles
all sides are the same lenght in a rhombus,
thus, find the area of all of the triangles,
and you already have two sides given,
and then, add them up
or just find the length of one, and multiply times 4 :)
recall, area of a triangle = 1/2 bh
<span>(3.5, 3) is the circumcenter of triangle ABC.
The circumcenter of a triangle is the intersection of the perpendicular bisectors of each side. All three of these perpendicular bisectors will intersect at the same point. So you have a nice self check to make sure your math is correct. Now let's calculate the equation for these bisectors.
Line segment AB:
Slope
(4-2)/(1-1) = 2/0 = infinity.
This line segment is perfectly vertical. So the bisector will be perfectly horizontal, and will pass through ((1+1)/2, (4+2)/2) = (2/2, 6/2) = (1,3).
So the equation for this perpendicular bisector is y = 3.
Line segment BC
(2-2)/(6-1) = 0/5 = 0
This line segment is perfectly horizontal. So the bisector will be perfectly vertical, and will pass through ((1+6)/2,(2+2)/2) = (7/2, 4/2) = (3.5, 2)
So the equation for this perpendicular bisector is x=3.5
So those two bisectors will intersect at point (3.5,3) which is the circumcenter of triangle ABC.
Now let's do a cross check to make sure that's correct.
Line segment AC
Slope = (4-2)/(1-6) = 2/-5 = -2/5
The perpendicular will have slope 5/2 = 2.5. So the equation is of the form
y = 2.5*x + b
And will pass through the point
((1+6)/2, (4+2)/2) = (7/2, 6/2) = (3.5, 3)
Plug in those coordinates and calculate b.
y = 2.5x + b
3 = 2.5*3.5 + b
3 = 8.75 + b
-5.75 = b
So the equation for the 3rd bisector is
y = 2.5x - 5.75
Now let's check if the intersection with this line against the other 2 works.
Determining intersection between bisector of AC and AB
y = 2.5x - 5.75
y = 3
3 = 2.5x - 5.75
8.75 = 2.5x
3.5 = x
And we get the correct value. Now to check AC and BC
y = 2.5x - 5.75
x = 3.5
y = 2.5*3.5 - 5.75
y = 8.75 - 5.75
y = 3
And we still get the correct intersection.</span>
<h3>
Answer: y = 10</h3>
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Work Shown:
EI = EG
3y-10 = 2y
3y-10-2y = 0
y-10 = 0
y = 0+10
y = 10
Note how if y = 10, then
- EI = 3y-10 = 3*10-10 = 20
- EG = 2y = 2*10 = 20
Both EI and EG are 20 units long when y = 10. This confirms the answer.
Answer:
3x³ + 23x² + 63x + 55
Step-by-step explanation:
Given
(3x + 5)(x² + 6x + 11)
Each term in the second factor is multiplied by each term in the first factor, that is
3x(x² + 6x + 11) + 5(x² + 6x + 11) ← distribute both parenthesis
= 3x³ + 18x² + 33x + 5x² + 30x + 55 ← collect like terms
= 3x³ + 23x² + 63x + 55
3 terms. Abcd is one term
E is another term. -h27 is the third term
