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# Trigonometry. Is this an ambiguous triangle?

(By the way, the line BE actually looks straight. It might not look straight in that picture)

Since 6cm<9cm, and if BE is a straight line, shouldn't AD=AC=6cm?

### 1 Answer

- RealProLv 73 years agoFavorite Answer
Actually no, it only looks like that.

It's true that triangle ABC as defined (if you didn't have a drawing, only numbers) is ambigous and you could move C to the right on BE resulting in a same length 6cm for AC somewhere, but there's nothing to say that location would be exactly D.

Drop a line straight down from A to point P, resulting in right triangle ABP.

AP = 9*sin(40°) = 5.79 cm

Now apply Pythagoras to ACP.

6^2 = 5.79^2 + CP^2

CP = 1.59 cm (this would have to be 4 to make AD = AC)

PD = 8 - CP = 6.41 cm

AD^2 = AP^2 + PD^2

AD = 8.63 cm <<<<<<<<<<<<<<<<<<<<<

For b) it's easiest to use formula for area of triangle

A = (1/2)(one side)(other side) * sin(angle between sides)

28 = (1/2)(AD)(AE)*sin(DAE)

This has two solutions but take the acute one, DAE = 40.43° <<<<<<<<<<<<<<<<<<<<

(Check my math)

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