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Geometry, NAP-F4-CA27 SA

Miko is designing a fan using similar triangles.
 

 
The edges of the largest and smallest triangles form a right angle.

What is the size of the angle marked `x^@`?

  degrees
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`75^@`

Show Worked Solution

`text(Consider the top angles of each triangle,)\ a°`

`6 xx a^@` `= 90^@`
`a^@` `= 90 ÷ 6`
  `= 15^@`
`:. x^@` `= 180 – (15 + 90)\ \ text{(angle sum of triangle)}`
  `= 75^@`

Geometry, NAP-A4-NC09

Two squares `ABCD` and `EFGH` are shown below.

 

 

The area of triangle `FGH` is

 
 quarter the area of `ABCD`
 
 four times the area of `ABCD`
 
 one eighth the area of `ABCD`
 
 eight times the area of `ABCD`
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`text(quarter the area of)\ ABCD`

Show Worked Solution
`text(Area)\ EFGH` `= 1/2\ text(Area)\ ABCD`
`text(Area)\ Delta FGH` `= 1/2\ text(Area)\ EFGH`
`:.\ text(Area)\ Delta FGH` `= 1/4\ text(Area)\ ABCD`

Geometry, NAP-A4-CA16

A regular pentagon, a square and an equilateral triangle meet at a point.
 

 
What is the size of the angle `x°`?

`112°` `110°` `108°` `102°`
 
 
 
 
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`102°`

Show Worked Solution

`text(Sum of internal angles of hexagon)`

`= (n-2) xx 180`

`= 3 xx 180`

`= 540^@`
 

`text(Internal angle in hexagon)`

`= 540/5`

`= 108^@`
 

`:. x` `= 360-(108 + 90 + 60)`
  `= 102^@`

Geometry, NAP-C4-CA31 SA

Euclid created the shape below using similar triangles.
 


 

What is the value of `x`?  
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`87^@`

Show Worked Solution

`text(S) text(ince there are 180° in a triangle,)`

`text(Centre angle)` `=180-(90+51)`
  `=39^@`

`text(The angles at the centre in each triangle will be)`

`text(the same because all traingles are similar.)`

`:. x^@` `= 360 – (7 xx 39)`
  `= 360 – 273`
  `= 87^@`

Geometry, NAP-C4-CA23 SA

Brett drew the shape below using rectangles and straight lines.

 

 

What is the value of  `x`?

   `text(degrees)`
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`164^@`

Show Worked Solution

`text(Using the right angle property of the rectangles,)`

`90^@ – 74^@ = 16^@`

`text(Vertically opposite angles are equal,)`

`:. x^@` `= 180 – 16`
  `= 164^@`

Geometry, NAP-E4-NC30 SA

Igor was designing a shield using 10 identical isosceles triangles, as shown in the diagram below.
 

How many degrees in the angle marked `x`?

  degrees
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`72^@`

Show Worked Solution

`text(Angles at centre of circle)\ = 360/10 = 36^@`

`text{Since triangles are isosceles:}`

`180` `= 36 + 2x`
`2x` `= 180-36`
  `= 144 `
`:. x` `= 72^@`

Geometry, NAP-G4-NC18

`AB` is the diameter of a circle, centre `O`.

There are 3 triangles drawn in the lower semi-circle and the angles at the centre are all equal to `x^@`.

The three triangles are best described as:

`text(isosceles)` `text(scalene)` `text(right-angled)` `text(equilateral)`
 
 
 
 
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`text(equilateral)`

Show Worked Solution
`3x` `=180`
`:. x` `=60^@`

 
`AO=OC=OD=OB\ \ text{(radii of circle)}`

`=>\ text{Since angles opposite equal sides of a triangle are}`

`text(equal, all triangle angles can be found to equal 60°.)`

`:.\ text(The three triangles are equilateral.)`