Question

In this figure, DC = 2.8, and BC = 9.3.

What is the length of AC?
Enter your answer rounded to the nearest tenth in the box.
AC =

Answer 1

Using the Pythagorean Theorem in right-angled triangles ABC and ABD, where DC = 2.8 and BC = 9.3, the length of AC is approximately 8.9 units, rounded to the nearest tenth.

Let's use the Pythagorean Theorem to find the length of AC.

1. Given: BA is perpendicular to AD, AC is perpendicular to DC and BC.

2. We have a right-angled triangle ABC, where AC is the hypotenuse.

3. Apply the Pythagorean Theorem:

`\[ AC^2 = AD^2 + DC^2 \]`

4. We also have another right-angled triangle ABD:

`\[ AB^2 = BC^2 + AC^2 \]`

5. Given values: DC = 2.8 and BC = 9.3.

6. Substitute these values into the Pythagorean Theorem equations:

`\[ AC^2 = AD^2 + (2.8)^2 \] \[ AB^2 = (9.3)^2 + AC^2 \]`

7. Rearrange the first equation to solve for
`\(AD^2\)`:

`\[ AD^2 = AC^2 - (2.8)^2 \]`

8. Substitute the expression for
`\(AD^2\)` into the second equation:

`\[ AB^2 = (9.3)^2 + (AC^2 - (2.8)^2) \]`

9. Simplify the equation:

`\[ AC^2 = √((9.3)^2 - (2.8)^2) \]`

10. Calculate the value of AC:

`\[ AC \approx √(86.49 - 7.84) \] \[ AC \approx √(78.65) \] \[ AC \approx 8.9 \]`

Therefore, the length of AC is approximately 8.9, rounded to the nearest tenth.