Hello math bugs(🐞) and hivers(🐝)
I hope you are doing good.
Today's article is about finding area when two sides and an angle between the sides of a triangle is given but no clue of its height. Try to find the area of the following triangle.
Yeah ,we can take help of trigonometry to find the height of a triangle. And thus we get a geo-trigo formulla to find area of a triangle as shown in the figure below:
You must have seen in the above figure AD is one of the heights of the triangle ∆ABC.From smaller triangle ∆ADC we can use trigonometry sine(sin) ratio to get value of perpendicular AD and thus we get one height of the given triangle. In that case our base will be BC.
To calculate the area we need to put value of height AD and base BC in the formulla. Value of BC is given in the problem and we find value of AD from sin ratio(AD/AC=sin30°).So, area is 1/2×BC×AC×sin30° sq. unit as shown in the figure below:
Points to be considered:
✅A triangle has three different heights according to on which base it is drawn from the opposite vertex.
✅To find area of a triangle from the above formulla the given angle must be between two given sides.
✅Between two given sides the required perpendicular can be drawn on any side.Actually you need not to draw any of them if you know how it works.
✅Do not get confused if the given triangle is an obtuse angle triangle.In that case the perpendicular must be on extended base.
✅This a generalise formulla and can be used on any
kind of triangle But the only thing you need to know you need two adjacent sides and angle between.
✅If the given angle is not an standard angle , don't worry we can use calculator to find the value of sin ratio.When the problem is asked in exam is a standard angle if it not an standard angle , we can use other formulla according to the info given.
Test: Find the third side of the ∆ABC. To do so you can use heron's formulla once you find the area of the triangle.
HERON'S FORMULLA:
Hint:Equate heron's formulla and the area you find.
There is only one side unknown.The equation will help you to find it.
I hope you find my article helpful.
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Have a nice day.
