CPM Homework Help : CCG Problem 5-135

Triangle A, B, C. Angle A is 45 degrees and angle C is 30 degrees. Two internal triangles are created by a dashed line segment of 4 drawn from the upper vertex perpendicular to the base. Find the area and perimeter of $ΔABC$ at right. Give approximate (decimal) answers, not exact answers.

Hint:

Separate $ΔABC$ into two separate triangles by dropping the height from vertex $B$ .
Label the new point $D$ , creating $ΔABD$ and $ΔDBC$ .

Point added, labeled, d, where the vertical line segment, intersects side, A, C, triangle, A, B, D, is shaded blue, & right triangle, B, C, D, is shaded red.

Help:

Both of the triangles that have been created are special right triangles, because the height $BD$ creates two right triangles: a $45º\text{-}45º\text{-}90º$ triangle and a $30º\text{-}60º\text{-}90º$ triangle.

More Help:

The length of the hypotenuse in a $45º\text{-}45º\text{-}90º$ triangle is the length of a leg times $\sqrt{2}$ .
In a $30º\text{-}60º\text{-}90º$ triangle, the length of the side opposite the $30^\circ$ angle is half of the length of the hypotenuse and the length of the side opposite the $60^\circ$ angle is half of the length of the hypotenuse times $\sqrt{3}$ .