A triangle has two corners with angles of $\frac{π}{2}$ $( pi ) / 2$ and $\frac{π}{6}$ $( pi )/ 6$ . If one side of the triangle has a length of $5$ $5$ , what is the largest possible area of the triangle?

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A triangle has two corners with angles of $\frac{π}{2}$ $( pi ) / 2$ and $\frac{π}{6}$ $( pi )/ 6$ . If one side of the triangle has a length of $5$ $5$ , what is the largest possible area of the triangle?

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Answer 1 · 2017-02-27T07:45:09

The largest possible area is $21.65$ $21.65$ .

Explanation:

As the two angles are $\frac{π}{2}$ $pi/2$ and $\frac{π}{6}$ $pi/6$ , the third angle is

$π - \frac{π}{2} - \frac{π}{6} = \frac{6 π}{6} - \frac{3 π}{6} - \frac{π}{6} = \frac{6 π - 3 π - π}{6} = \frac{2 π}{6} = \frac{π}{3}$ $pi-pi/2-pi/6=(6pi)/6-(3pi)/6-pi/6=(6pi-3pi-pi)/6=(2pi)/6=pi/3$

As the smallest angle of the three is $\frac{π}{6}$ $pi/6$ ,

area will be largest if side with length $5$ $5$ is opposite the smallest angle $\frac{π}{6}$ $pi/6$

Such a triangle is typical $30^{\circ} - 60^{\circ} - 90^{\circ}$ $30^@-60^@-90^@$ triangle and looks like as shown below

enter image source here

In such a right angled triangle, if smallest side is $5$ $5$ , hypotenuse is $5 \times 2 = 10$ $5xx2=10$ and third side $5 \times \sqrt{3}$ $5xxsqrt3$ .

As the two sides forming the right angle in right angled triangle are $5$ $5$ and $5 \sqrt{3}$ $5sqrt3$ , the area of triangle is

$\frac{1}{2} \times 5 \times 5 \sqrt{3} = \frac{25}{2} \times 1.732 = 21.65$ $1/2xx5xx5sqrt3=25/2xx1.732=21.65$ and

The largest possible area is $21.65$ $21.65$ .

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A triangle has two corners with angles of $\frac{π}{2}$ $( pi ) / 2$ and $\frac{π}{6}$ $( pi )/ 6$ . If one side of the triangle has a length of $5$ $5$ , what is the largest possible area of the triangle?

1 Answer

Explanation:

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A triangle has two corners with angles of $\frac{π}{2}$ $( pi ) / 2$ and $\frac{π}{6}$ $( pi )/ 6$ . If one side of the triangle has a length of $5$ $5$ , what is the largest possible area of the triangle?