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ABCD is a rectangle tacking AD asA diameter a semicircle AXD is drawn which intersects diagonal BD. at X lf AB=12cm ,AD=9cm then find BD and BX?
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ABCD is a rectangle tacking AD asA diameter a semicircle AXD is drawn ...

Given:
- ABCD is a rectangle with AD as a diameter.
- A semicircle AXD is drawn which intersects diagonal BD at X.
- AB = 12 cm and AD = 9 cm.

To find:
- BD and BX.

Solution:
We can start by finding the length of BD using the Pythagorean theorem.

Step 1: Finding BD
In rectangle ABCD, we can see that AD is the diameter of the semicircle AXD. Therefore, the radius of the semicircle is half the length of AD, which is 9/2 = 4.5 cm.

In triangle ABX, we have a right triangle with AB as the hypotenuse and BX as one of the legs. The other leg is AX, which is the radius of the semicircle.

Using the Pythagorean theorem, we can write:
AB² = AX² + BX²

Substituting the given values:
12² = (4.5)² + BX²
144 = 20.25 + BX²
BX² = 144 - 20.25
BX² = 123.75
BX = √123.75
BX ≈ 11.11 cm

So, the length of BX is approximately 11.11 cm.

Step 2: Finding BD using Similar Triangles
Now, we can find BD by using the fact that triangle BDX is similar to triangle ABX.

The ratio of corresponding sides in similar triangles is equal. Therefore, we can write:
BD/BX = AD/AB

Substituting the given values:
BD/11.11 = 9/12

Cross-multiplying, we get:
BD = (9 * 11.11) / 12
BD ≈ 8.33 cm

So, the length of BD is approximately 8.33 cm.

Summary:
- The length of BX is approximately 11.11 cm.
- The length of BD is approximately 8.33 cm.
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ABCD is a rectangle tacking AD asA diameter a semicircle AXD is drawn which intersects diagonal BD. at X lf AB=12cm ,AD=9cm then find BD and BX?
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