XY is a diameter of a circle and Z is a point on the circle such that ZY=6. If the area of the triangle XYZ is 18 square root 3 find the length of arc XZ

4π

As shown in the diagram, triangle XYZ is a right triangle. Therefore, its area (A) is given by:

A = x b x h      -------------(i)

Where;

A = 18

b = XZ = base of the triangle

h = YZ = height of the triangle = 6

Substitute these values into equation(i) and solve as follows:

18 =   x b x 6

18 =  3b

Divide through by 3

6 =  b

Therefore, b = XZ = 6

Now, assume that the circle is centered at O;

Triangle XOZ is isosceles, therefore the following are true;

(i) |OZ| = |OX|

(ii) XZO = ZXO = 30°

(iii) XOZ + XZO + ZXO = 180°   [sum of angles in a triangle]

=>  XOZ + 30° + 30° = 180°

=>  XOZ + 60° = 180°

=>  XOZ = 180° - 60°

=>  XOZ = 120°

Therefore we can calculate the radius |OZ| of the circle using sine rule as follows;

|OZ| = 6

The radius of the circle is therefore 6.

Now, let's calculate the length of the arc XZ

The length(L) of an arc is given by;

L = θ / 360 x 2 π r          ------------------(ii)

Where;

θ = angle subtended by the arc at the center.

r = radius of the circle.

In our case,

θ = ZOX = 120°

r = |OZ| = 6

Substitute these values into equation (ii) as follows;

L = 120/360 x 2π x 6

L = 4π

Therefore the length of the arc XZ is 4π

(0,,,0)

step-by-step explanation:

the table of x and y values is shown in the attached pic(part of question)

we have to decide according that graph

given points are (0, 0) (–1, 0), (2, 0)

when we graph these points on graph they show an increasing line as shown in the attached pic