If m<DBC=10x and m<ACb=4x^2, find m<ACB.
The quadrilateral ABCD is a rectangle.
B ------------- C
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A -------------|D
There are diagonal bisectors inside the rectangle but I could not draw them in. The diagonals are DB and CA and the point in the middle is E.
Let E be where the diagonals intersect.
Since ABCD is a rectangle, the diagonals are the same length, and bisect each other.
Thus, EB = EC and the triangle BCE is isosceles, making m<DBC = m<ACB
So,
10x = 4x^2
10 = 4x
x = 5/2 = 2.5
<DBC = 25°
<ACB = 25°
the coordinates of the vertices of abc are a (2 5) B (6, -1) and C (-4,-2). FInd the perimeter of abc, to the nearest tenth
To find the measure of angle ACB, we need to consider the given measurements m<DBC and m<ACb.
First, let's consider the diagonal DB. Since angle DBC is given as 10x and angle CBE is a bisector, we can set them equal to each other:
m<DBC = m<CBE
10x = m<CBE ----- (Equation 1)
Next, let's consider the diagonal CA. Since angle ACb is given as 4x^2 and angle CBE is a bisector, we can set them equal to each other:
m<ACb = m<CBE
4x^2 = m<CBE ----- (Equation 2)
Now, we have two equations (Equation 1 and Equation 2) with one unknown (m<CBE). We can solve these equations simultaneously to find the value of x.
From Equation 1:
10x = m<CBE
From Equation 2:
4x^2 = m<CBE
Now, we can set these two equations equal to each other:
10x = 4x^2
Rearranging the equation:
4x^2 - 10x = 0
Factoring out x:
2x(2x - 5) = 0
So, either 2x = 0 or 2x - 5 = 0.
If 2x = 0, then x = 0.
If 2x - 5 = 0, then 2x = 5, giving x = 5/2.
Now, we have found the possible values for x, which are x = 0 and x = 5/2.
Next, substitute these values back into either Equation 1 or Equation 2 to find the corresponding measure of angle CBE.
For x = 0:
m<CBE = 10x = 10(0) = 0
For x = 5/2:
m<CBE = 4x^2 = 4(5/2)^2 = 4(25/4) = 25
Therefore, angle CBE can be either 0 or 25 degrees.
Now, let's find the measure of angle ACB by adding the measures of angle CBE and angle ACb:
m<ACB = m<CBE + m<ACb
If m<CBE = 0:
m<ACB = 0 + m<ACb = 0 + 4x^2 = 4x^2
If m<CBE = 25:
m<ACB = 25 + m<ACb = 25 + 4x^2
Therefore, the measure of angle ACB can be either 4x^2 or 25 + 4x^2, depending on the value of x.