Math - jose, Sunday, October 14, 2007 at 12:55pm
<EAB and < ABC are right angles,AB=4,BC=6,AE=8,AC and BE interset at D.What is the differens the area of triangle ADE and triangle BDC
To find the difference in the area of triangle ADE and triangle BDC, we first need to find the areas of both triangles.
To find the area of triangle ADE, we'll use the formula for the area of a triangle:
Area = (1/2) * base * height
First, we need to find the base and height of triangle ADE. Since <EAB and <ABC are right angles, we can use the Pythagorean Theorem to find the lengths of AD and DE.
Using the Pythagorean Theorem:
AD^2 = AE^2 - DE^2
AD^2 = 8^2 - DE^2
AD^2 = 64 - DE^2
Similarly, using the Pythagorean Theorem:
DE^2 = BC^2 - BE^2
DE^2 = 6^2 - 4^2
DE^2 = 36 - 16
DE^2 = 20
Substituting the value of DE^2 into the equation for AD^2:
AD^2 = 64 - 20
AD^2 = 44
Taking the square root of both sides:
AD = √44
AD ≈ 6.63
Now that we have the base (AD) and height (DE) of triangle ADE, we can calculate its area:
Area of ADE = (1/2) * AD * DE
Area of ADE = (1/2) * 6.63 * √20
Area of ADE ≈ 16.47
Next, we need to find the area of triangle BDC. Since triangle BDC is similar to triangle ADE, they have the same shape but different sizes. Since the scale factor between the two triangles is 2, the ratio of their areas is 2^2 = 4. Therefore, the area of triangle BDC is four times smaller than the area of triangle ADE.
Area of BDC = (1/4) * Area of ADE
Area of BDC = (1/4) * 16.47
Area of BDC ≈ 4.12
Finally, we can calculate the difference in the area of triangle ADE and triangle BDC:
Difference in area = Area of ADE - Area of BDC
Difference in area = 16.47 - 4.12
Difference in area ≈ 12.35
Therefore, the difference in the area of triangle ADE and triangle BDC is approximately 12.35 units.