Right triangle ABC has legs AB = 6 cm and AC = 8 cm. Square BCDE is drawn
outside of the triangle along the hypotenuse. What is the area of triangle ADE?
To find the area of triangle ADE, we need to first find the lengths of its sides.
Since triangle ABC is a right triangle, we can use the Pythagorean theorem to find the length of BC.
According to the theorem:
BC^2 = AB^2 + AC^2
BC^2 = 6^2 + 8^2
BC^2 = 36 + 64
BC^2 = 100
BC = √100
BC = 10 cm
Since BCDE is a square and BC is one of its sides, all the other sides of the square are also equal to BC. Therefore, DE = 10 cm.
To find the area of triangle ADE, we need the height of the triangle. The height is the length of the perpendicular line segment from vertex A to side DE.
Since triangle ABC is a right triangle, the altitude from right angle B divides the triangle into two smaller triangles, ABD and ACD, which are similar to the original triangle ABC.
The ratio of the sides of these similar triangles is:
AB/BD = AC/CD
6/BD = 8/(BC - BD)
6/BD = 8/(10 - BD)
6(10 - BD) = 8BD
60 - 6BD = 8BD
60 = 14BD
BD = 60/14
BD ≈ 4.286 cm
The height of triangle ADE is BD, which is approximately 4.286 cm.
Now we can calculate the area of triangle ADE using the formula:
Area = (1/2) * base * height
Area = (1/2) * DE * BD
Area = (1/2) * 10 * 4.286
Area ≈ 21.43 cm²
Therefore, the area of triangle ADE is approximately 21.43 square centimeters.