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?

1 answer

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.
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