smc-3589-30-Heron’s formula

GEOMETRY, FUR1 2021 VCAA 4 MC

The side length of an equilateral triangle is 4 cm, as shown in the diagram below.

Which one of the following is not a correct calculation for the area of this triangle?

`{sqrt3 xx 4^2}/{4}`
`2 xx sqrt12`
`1/2 xx 4 xx 4`
`4^2/2 xx sin(60^@)`
`sqrt{6 (6-4)^3}`

Show Answers Only

`C`

Show Worked Solution

`text{By Pythagoras:}`

`h = sqrt{4^2 -2^2} = sqrt12 = 2 sqrt3`

`text{Area}`	`= 2 xx (1/2 xx 2 xx 2 sqrt3)`
	`= 4 sqrt3 \ text{cm}^2`

`text{Option C is the only option} ≠ 4 sqrt3`

`=> C`

GEOMETRY, FUR1 2006 VCAA 5 MC

A block of land is triangular in shape.

The three sides measure 36 m, 58 m and 42 m.

To calculate the area, Heron’s formula is used.

The correct application of Heron’s formula for this triangle is

`text(Area) = sqrt(136\ (136 − 36) (136 − 58) (136 − 42))`
`text(Area) =sqrt(136\ (136 −18) (136 − 29) (136 − 21))`
`text(Area) =sqrt(68\ (68 − 36) (68 − 58) (68 − 42))`
`text(Area) = sqrt(68\ (68 −18) (68 − 29) (68 − 21))`
`text(Area) = sqrt(68\ (136 − 36) (136 − 58) (136 − 42))`

Show Answers Only

`C`

Show Worked Solution

`s`	`= (36 + 58 + 42)/2`
	`= 68`

`:.\ text(Area)`	`= sqrt(s(s – a) xx (s – b) xx (s – c))`
	`= sqrt(68 (68 – 36) xx (68 – 58) xx (68 – 42))`

`=> C`

GEOMETRY, FUR1 2013 VCAA 2 MC

The distances from a kiosk to points `A` and `B` on opposite sides of a pond are found to be 12.6 m and 19.2 m respectively.

The angle between the lines joining these points to the kiosk is 63°.

The distance, in m, across the pond between points `A` and `B` can be found by evaluating

A. `1/2 xx 12.6 xx 19.2 xx sin(63°)`

B. `{19.2 xx sin(63°)}/12.6`

C. `sqrt(12.6^2 + 19.2^2)`

D. `sqrt(12.6^2 + 19.2^2 - 2 xx 12.6 xx 19.2 xx cos(63°)`

E. `sqrt{s(s - 12.6)(s - 19.2)(s - 63)} , text(where)\ s = 1/2 (12.6 + 10.2 + 63)`

Show Answers Only

`D`

Show Worked Solution

`text(Using the cosine rule:)`

`(AB)^2`	`= 12.6^2 xx 19.2^2 – 2 xx 12.6 xx 19.2 xx cos(63°)`
`:. AB`	`= sqrt{12.6^2 xx 19.2^2 – 2 xx 12.6 xx 19.2 xx cos(63°)}`

`=>D`

GEOMETRY, FUR1 2009 VCAA 2 MC

The area (in m²) of triangle `XYZ` can be found using Heron’s formula `A = sqrt(s(s−a)(s−b)(s−c))`, with `a = 1.92`, `b = 8.24`, `c = 9.20` and `s =`

A. `4.40`

B. `6.45`

C. `9.20`

D. `9.68`

E. `19.36`

Show Answers Only

`D`

Show Worked Solution

`s`	`= (a + b + c) / 2`
	`= (1.92 + 8.24 + 9.20) /2`
	`= 9.68`

`=> D`