Question

At the base triangle ABC is the rectangle BCDE, CD = 3 cm, AC = 12 cm and

∠ = 300.Find the length of AE and DE.​

Answer 1

Answer:

Length of AE = 6cm

Length of DE = 3

Step-by-step explanation:

To solve this problem, we will need to use the properties of special triangles and the Pythagorean theorem. Here are the steps:

• Since angle BAC is 60 degrees, triangle ABC is an equilateral triangle, which means all sides are equal. Let's call this length x.

• Since CD = 3 cm, we can use the Pythagorean theorem to find the length of BD. We know that BD^2 = BC^2 - CD^2, so BD^2 = x^2 - 9.

• Since rectangle BCDE has a right angle at point D, we know that triangle ADE is also a right triangle. Therefore, we can use the Pythagorean theorem to find the length of AE. We know that AE^2 = AD^2 + DE^2, so AE^2 = (x + 3)^2 + BD^2.

• We can substitute the value of BD^2 from step 2 into the equation in step 3: AE^2 = (x + 3)^2 + x^2 - 9.

• We can simplify this equation to get AE^2 = 2x^2 + 6x.

• We also know that AC = 12 cm, which is equal to x + 3 (since AC = AB + BC = x + x = 2x, and AB = x and BC = x). Therefore, we can solve for x:

x + 3 = 12

x = 9

• Now that we know x, we can find BD^2:

BD^2 = x^2 - 9 = 72

BD = sqrt(72) = 6sqrt(2)

• Finally, we can use x and BD to find AE:

AE^2 = 2x^2 + 6x = 2(9^2) + 6(9) = 162 + 54 = 216

AE = sqrt(216) = 6sqrt(6)

Therefore, the length of AE is 6sqrt(6) cm and the length of DE is 3 cm.