in the diagram AB=20 cm, BC=21 cm, AD=10 cm and DE= 10.5 cm. Angles ABC and ADE are right angles. if triangleADE is removed from triangleABC.What is the area of shaded region that remains?

To find the area of the shaded region, we need to subtract the area of triangle ADE from the area of triangle ABC.

Step 1: Calculate the area of triangle ABC.
Since we are given the lengths of AB and BC, we can use the formula for the area of a triangle: area = 1/2 * base * height.
The base of triangle ABC is BC, which is 21 cm.
The height of triangle ABC can be found by drawing a perpendicular line from B to AC. Let's call the point where the perpendicular line intersects AC as F.
Using triangle BFC, we can find the height BF. We have BF = BC - CF.
Since angle ABC is a right angle and triangle BFC is also a right triangle, we can use the Pythagorean theorem to find CF.
CF^2 = BC^2 - BF^2
CF^2 = 21^2 - 20^2
CF^2 = 441 - 400
CF^2 = 41
CF = √41 ≈ 6.4 cm
BF = BC - CF = 21 - 6.4 ≈ 14.6 cm

Now we have the base and height values for triangle ABC, we can calculate its area.
Area_ABC = 1/2 * BC * BF = 1/2 * 21 * 14.6 ≈ 153.3 cm²

Step 2: Calculate the area of triangle ADE.
Triangle ADE is a right triangle, and we are given the lengths of AD and DE.
Area_ADE = 1/2 * AD * DE = 1/2 * 10 * 10.5 = 52.5 cm²

Step 3: Subtract the area of triangle ADE from the area of triangle ABC to find the shaded region.
Area_shaded_region = Area_ABC - Area_ADE = 153.3 - 52.5 = 100.8 cm²

Therefore, the area of the shaded region that remains is approximately 100.8 cm².

Missing information: presumably A, D, B are colinear.

For right angle XYZ, |∆XYZ|=(1/2)(XY×YX)

So:
|∆ABC|-|∆ADE| = (1/2)(AB×BC)-(1/2)(AD×DE)