Question

What is the length of segment AE?

(3 points – 1 point correct formula(s), 1 point correct proportion(s), 1 point accurate calculations)

Answer 1

`\text{AE} = (50)/(3)`.

Explanation:

Step 1: Show that triangle ABC is similar to triangle CDE.

Segment BC is parallel to segment DE. As a result,

`\angle \text{BCD} = \angle \text{CDE} = 90\textdegree{}`.

Point A, C, and E lines up on segment AE. As a result,
`\angle \text{ACE} = 180 \textdegree{}`.

`\angle \text{ACB} + \angle \text{BCD} + \angle \text{DCE} = \angle \text{ACE} = 180 \textdegree{}`.

As seen above,
`\angle \text{BCD} = 90 \textdegree{}`. Therefore,

`\angle \text{ACB} +90\textdegree{}+ \angle \text{DCE} = \angle \text{ACE} = 180 \textdegree{}\\\angle \text{ACB} +\angle \text{DCE} = 90 \textdegree{}`.

The three angles of triangle ABC adds up to
`180 \textdegree{}`. That is:

`\angle\text{BAC} + \angle \text{ACB}+ \angle \text{ABC} = 180 \textdegree{}`.

`\angle\text{ABC} = 90\textdegree{}` for being a right angle. As a result,

`\angle\text{BAC} + \angle \text{ACB}+ 90 \textdegree{} = 180 \textdegree{}\\\angle\text{BAC} + \angle \text{ACB} = 90 \textdegree{}`

`\angle\text{BAC} + \angle \text{ACB} = 90 \textdegree{} = \angle \text{ACB} +\angle \text{DCE} \\\angle\text{BAC} + \angle \text{ACB} = \angle \text{DCE} + \angle \text{ACB} \\\angle\text{BAC}= \angle \text{DCE}`

Two of triangle ABC and CDE's angles are the same, implying that the third angle is also the same. As a result, the two triangles are similar.

Step 2: Find the ratios between the segments.

Only the length of CD is known among all three sides in triangle CDE. Side CE is part of segment AE; the length of side CE needs to be determined. Side CD corresponds to side AB, whereas side CE corresponds to side AC.

`\frac{\text{AC}}{\text{AB}} = \frac{\text{CE}}{\text{CD}}`.

Step 3: Find the length of AC, CE, and hence AE.

Apply the Pythagorean Theorem:

`\text{AC} = \sqrt{\text{AB}^(2) + \text{BC}^(2)}\\\phantom{\text{AC}} = √(6^2 + 8^2)\\\phantom{\text{AC}} = 10`

Apply the ratio found in step 2:

`\text{AC} = \text{AB} \cdot \frac{\text{CE}}{\text{CD}}\\\phantom{\text{AC}} = 10 * (4)/(6)\\\phantom{\text{AC}} = (20)/(3)`.

`\text{AE} = \text{AC} + \text{CE} = 10 + (20)/(3) = (50)/(3)`.