The questions are illustrations of Pythagoras theorem and trigonometry ratios.
<h3>What is Pythagoras theorem?</h3>
Pythagoras theorem is used to determine the lengths of the legs of a right triangle.
It is represented as:
![a^2 = b^2 + c^2](https://tex.z-dn.net/?f=a%5E2%20%3D%20b%5E2%20%2B%20c%5E2)
Where:
b & c are the legs of the triangle, while a is the hypotenuse
<u>Question 16</u>
Start by calculating the value of x, using the sine ratio
![\sin(30) = \frac{x}{16\sqrt 3}](https://tex.z-dn.net/?f=%5Csin%2830%29%20%3D%20%5Cfrac%7Bx%7D%7B16%5Csqrt%203%7D)
Evaluate sin(30)
![0.5 = \frac{x}{16\sqrt 3}](https://tex.z-dn.net/?f=0.5%20%3D%20%5Cfrac%7Bx%7D%7B16%5Csqrt%203%7D)
Solve for x
![x = 0.5 * 16\sqrt 3](https://tex.z-dn.net/?f=x%20%3D%200.5%20%2A%2016%5Csqrt%203)
![x = 8\sqrt 3](https://tex.z-dn.net/?f=x%20%3D%208%5Csqrt%203)
Next, calculate the length (l) of the boundary between both triangles using Pythagoras theorem
![(16\sqrt 3)^2 = (8\sqrt 3)^2 +l^2](https://tex.z-dn.net/?f=%2816%5Csqrt%203%29%5E2%20%3D%20%288%5Csqrt%203%29%5E2%20%2Bl%5E2)
![768 = 192 +l^2](https://tex.z-dn.net/?f=768%20%3D%20192%20%2Bl%5E2)
Collect like terms
![l^2 = 768 -192](https://tex.z-dn.net/?f=l%5E2%20%3D%20768%20-192)
![l^2 = 576](https://tex.z-dn.net/?f=l%5E2%20%3D%20576)
Take the square root of both sides
![l = 24](https://tex.z-dn.net/?f=l%20%3D%2024)
Given that the angle is 45 degrees, it means that:
![z = y](https://tex.z-dn.net/?f=z%20%3D%20y)
So, we have:
![z^2 + y^2 = 24^2](https://tex.z-dn.net/?f=z%5E2%20%2B%20y%5E2%20%3D%2024%5E2)
![z^2 + z^2 = 24^2](https://tex.z-dn.net/?f=z%5E2%20%2B%20z%5E2%20%3D%2024%5E2)
![2z^2 = 576](https://tex.z-dn.net/?f=2z%5E2%20%3D%20576)
Divide through by 2
![z^2 = 288](https://tex.z-dn.net/?f=z%5E2%20%3D%20288)
Take the square root of both sides
![z= 12\sqrt 2](https://tex.z-dn.net/?f=z%3D%2012%5Csqrt%202)
Hence, the values of x, y and z are:
![x = 8\sqrt 3](https://tex.z-dn.net/?f=x%20%3D%208%5Csqrt%203)
![y= 12\sqrt 2](https://tex.z-dn.net/?f=y%3D%2012%5Csqrt%202)
![z= 12\sqrt 2](https://tex.z-dn.net/?f=z%3D%2012%5Csqrt%202)
<u>Question 18</u>
Start by calculating the value of z, using the sine ratio
![\sin(45) = \frac{z}{20}](https://tex.z-dn.net/?f=%5Csin%2845%29%20%3D%20%5Cfrac%7Bz%7D%7B20%7D)
Evaluate sin(45)
![\frac{\sqrt 2}{2} = \frac{z}{20}](https://tex.z-dn.net/?f=%5Cfrac%7B%5Csqrt%202%7D%7B2%7D%20%3D%20%5Cfrac%7Bz%7D%7B20%7D)
Solve for z
![z = \frac{\sqrt 2}{2} * 20](https://tex.z-dn.net/?f=z%20%3D%20%5Cfrac%7B%5Csqrt%202%7D%7B2%7D%20%2A%2020)
![z = 10\sqrt 2](https://tex.z-dn.net/?f=z%20%3D%2010%5Csqrt%202)
Next, calculate the value of y using tangent ratio
![\tan(30) = \frac{y}{10\sqrt 2}](https://tex.z-dn.net/?f=%5Ctan%2830%29%20%3D%20%5Cfrac%7By%7D%7B10%5Csqrt%202%7D)
Solve for y
![y = \tan(30) * 10\sqrt 2](https://tex.z-dn.net/?f=y%20%3D%20%5Ctan%2830%29%20%2A%2010%5Csqrt%202)
Evaluate tan(30)
![y = \frac{\sqrt 3}{3} * 10\sqrt 2](https://tex.z-dn.net/?f=y%20%3D%20%5Cfrac%7B%5Csqrt%203%7D%7B3%7D%20%2A%2010%5Csqrt%202)
![y = \frac{10\sqrt 6}{3}](https://tex.z-dn.net/?f=y%20%3D%20%5Cfrac%7B10%5Csqrt%206%7D%7B3%7D)
Next, calculate the value of x using sine ratio
![\sin(30) = \frac{10/3\sqrt 6}{x}](https://tex.z-dn.net/?f=%5Csin%2830%29%20%3D%20%5Cfrac%7B10%2F3%5Csqrt%206%7D%7Bx%7D)
Solve for x
![x = \frac{10/3\sqrt 6}{\sin(30)}](https://tex.z-dn.net/?f=x%20%3D%20%5Cfrac%7B10%2F3%5Csqrt%206%7D%7B%5Csin%2830%29%7D)
Evaluate sin(30)
![x = \frac{10/3\sqrt 6}{1/2}](https://tex.z-dn.net/?f=x%20%3D%20%5Cfrac%7B10%2F3%5Csqrt%206%7D%7B1%2F2%7D)
![x = \frac{20\sqrt 6}{3}](https://tex.z-dn.net/?f=x%20%3D%20%5Cfrac%7B20%5Csqrt%206%7D%7B3%7D)
Hence, the values of x, y and z are:
![x = \frac{20\sqrt 6}{3}](https://tex.z-dn.net/?f=x%20%3D%20%5Cfrac%7B20%5Csqrt%206%7D%7B3%7D)
![y = \frac{10\sqrt 6}{3}](https://tex.z-dn.net/?f=y%20%3D%20%5Cfrac%7B10%5Csqrt%206%7D%7B3%7D)
![z = 10\sqrt 2](https://tex.z-dn.net/?f=z%20%3D%2010%5Csqrt%202)
<u>Question 20</u>
Start by calculating the value of z, using the sine ratio
![\sin(45) = \frac{z}{10\sqrt 6}](https://tex.z-dn.net/?f=%5Csin%2845%29%20%3D%20%5Cfrac%7Bz%7D%7B10%5Csqrt%206%7D)
Evaluate sin(45)
![\frac{\sqrt 2}{2} = \frac{z}{10\sqrt 6}](https://tex.z-dn.net/?f=%5Cfrac%7B%5Csqrt%202%7D%7B2%7D%20%3D%20%5Cfrac%7Bz%7D%7B10%5Csqrt%206%7D)
Solve for z
![z = \frac{\sqrt 2}{2} * 10\sqrt 6](https://tex.z-dn.net/?f=z%20%3D%20%5Cfrac%7B%5Csqrt%202%7D%7B2%7D%20%2A%2010%5Csqrt%206)
![z = 5\sqrt {12](https://tex.z-dn.net/?f=z%20%3D%205%5Csqrt%20%7B12)
Simplify
![z = 10\sqrt {3](https://tex.z-dn.net/?f=z%20%3D%2010%5Csqrt%20%7B3)
Next, calculate the value of y using tangent ratio
![\tan(30) = \frac{10\sqrt 3}{y}](https://tex.z-dn.net/?f=%5Ctan%2830%29%20%3D%20%5Cfrac%7B10%5Csqrt%203%7D%7By%7D)
Solve for y
![y = \frac{10\sqrt 3}{\tan(30)}](https://tex.z-dn.net/?f=y%20%3D%20%5Cfrac%7B10%5Csqrt%203%7D%7B%5Ctan%2830%29%7D)
Evaluate tan(30)
![y = \frac{10\sqrt 3}{1/\sqrt 3}](https://tex.z-dn.net/?f=y%20%3D%20%5Cfrac%7B10%5Csqrt%203%7D%7B1%2F%5Csqrt%203%7D)
Simplify
![y = 10\sqrt 3 * \sqrt 3](https://tex.z-dn.net/?f=y%20%3D%2010%5Csqrt%203%20%2A%20%5Csqrt%203)
![y = 30](https://tex.z-dn.net/?f=y%20%3D%2030)
Next, calculate the value of x using sine ratio
![\sin(30) = \frac{10\sqrt 3}{x}](https://tex.z-dn.net/?f=%5Csin%2830%29%20%3D%20%5Cfrac%7B10%5Csqrt%203%7D%7Bx%7D)
Solve for x
![x = \frac{10\sqrt 3}{\sin(30)}](https://tex.z-dn.net/?f=x%20%3D%20%5Cfrac%7B10%5Csqrt%203%7D%7B%5Csin%2830%29%7D)
Evaluate sin(30)
![x = \frac{10\sqrt 3}{1/2}](https://tex.z-dn.net/?f=x%20%3D%20%5Cfrac%7B10%5Csqrt%203%7D%7B1%2F2%7D)
![x = 20\sqrt 3](https://tex.z-dn.net/?f=x%20%3D%2020%5Csqrt%203)
Hence, the values of x, y and z are:
![x = 20\sqrt 3](https://tex.z-dn.net/?f=x%20%3D%2020%5Csqrt%203)
![y = 30](https://tex.z-dn.net/?f=y%20%3D%2030)
![z = 10\sqrt {3](https://tex.z-dn.net/?f=z%20%3D%2010%5Csqrt%20%7B3)
Read more about Pythagoras theorem and trigonometry ratios at:
brainly.com/question/6241673