Question

In the figure below, ∠ ABC ≅ ∠ DFE, ∠ BAC ≅ ∠ FDE, D and F are on AB, AD ≅ FB, and distances in centimeters are as shown. What is the length of AD, in centimeters?

Answer 1

The length of AD is 10 centimeters.

It seems like you're describing a scenario involving two triangles, ABC and DFE, where triangle DFE is inscribed in triangle ABC. You've mentioned that ∠ABC ≅ ∠DFE, ∠BAC ≅ ∠FDE, D and F are on AB, AD ≅ FB, FD = 6, AC = 20, and DE = 12.

Given that ∠ABC ≅ ∠DFE and ∠BAC ≅ ∠FDE, this implies that triangles ABC and DFE are similar by the Angle-Angle (AA) similarity criterion.

Since triangle DFE is inscribed in triangle ABC, we can also use the Inscribed Angle Theorem, which states that an angle inscribed in a semicircle is a right angle. Therefore, ∠DFE is a right angle.

Now, looking at triangles ABC and DFE:

1. ∠ABC ≅ ∠DFE (Given)

2. ∠BAC ≅ ∠FDE (Given)

3. ∠ACB is a right angle (Because AC is the diameter of the circle, and ∠ABC is inscribed in a semicircle, so it's a right angle).

This implies that triangles ABC and DFE are similar by AA, and triangle ABC is a right-angled triangle.

Now, let's consider the ratio of corresponding sides of the two similar triangles:

`\[(AD)/(DF) = (AC)/(DE)\]`

Substitute the given values:

`\[(AD)/(6) = (20)/(12)\]`

Now, solve for AD:

`\[AD = (20 * 6)/(12) = 10\]`

So, the length of AD is 10 centimeters.