# The sail on a toy boat is in the shape of an isosceles triangle. Its legs are 12 inches long and its base angles measure 70 degrees. How long to the nearest hundredth of an inch, is the base of the sail?

Nov 20, 2015

$8.21$"

#### Explanation:

Draw the triangle.

If both base angles are 70˚, and the sum of all the angles in a triangle is 180˚, we can determine that the other angles measure is x˚, where $70 + 70 + x = 180$, so $x = 40$.

If we have a 40˚-70˚-70˚ triangle, we can draw in its height from the top point to the midpoint of the base, creating two congruent right triangles within the isosceles triangle. In doing so, the 40˚ angle at the top of the triangle is bisected, splitting it into two 20˚ angles on either side of the triangle's altitude.

So, we now have two right triangles. Their hypotenuses are the legs of the isosceles triangle with length $12$, and we want to determine the length of the side opposite the 20˚ angle.

We should know that the $\text{cosine}$ of an angle in a right triangle is equal to the adjacent leg, which is $\frac{1}{2}$ the base of the isosceles triangle, divided by the hypotenuse, which has length $12$.

Therefore, we know that cos70˚=x/12, where $x$ is $\frac{1}{2}$ the base of the sail.

Through algebraic manipulation, we know that x=12cos70˚=4.10.

Since this is only $\frac{1}{2}$ the base, the base's entire length is $8.21$".

(Note that the $4.10$ value was rounded down, but when multiplied by $2$ had to be rounded up.)