Photo AI

[AB] and [CD] are chords of a circle that intersect externally at E, as shown - Leaving Cert Mathematics - Question 6B - 2014

Question 6B

[AB] and [CD] are chords of a circle that intersect externally at E, as shown. (a) Name two similar triangles in the diagram above and give reasons for your answer.... show full transcript

Worked Solution & Example Answer:[AB] and [CD] are chords of a circle that intersect externally at E, as shown - Leaving Cert Mathematics - Question 6B - 2014

Step 1

Name two similar triangles in the diagram above and give reasons for your answer.

96%

114 rated

Answer

The triangles ΔADE and ΔBCE are similar.

This is supported by the fact that
- ∠EAD = ∠EBC (angles subtended by the same arc BD)
- ∠ADE = ∠CEB (same angle)
- ∠DEA = ∠BCE (third angle)

Thus, the two triangles are similar by the Angle-Angle (AA) criterion.

Step 2

Prove that |EA| ∙ |EB| = |EC| ∙ |ED|.

99%

104 rated

Answer

Since triangles ΔADE and ΔBCE are similar, we can set up the following ratio:

$\frac{|EA|}{|EB|} = \frac{|EC|}{|ED|}$

By cross-multiplying, we get the equation: $|EA| ∙ |ED| = |EC| ∙ |EB|$

This proves that |EA| ∙ |EB| = |EC| ∙ |ED|.

Step 3

Given that |EB| = 6.25, |ED| = 5.94 and |CB| = 10, find |AD|.

96%

101 rated

Answer

Using the established proportion:

$\frac{|ED|}{|EB|} = \frac{|AD|}{|CB|}$

Substituting the known values: $\frac{5.94}{6.25} = \frac{|AD|}{10}$

Cross-multiplying gives: $|AD| = \frac{5.94 \cdot 10}{6.25}$

Calculating this, we find: $|AD| \approx 9.504$

Join the Leaving Cert students using SimpleStudy...

97% of Students

Report Improved Results

98% of Students

Recommend to friends

100,000+

Students Supported

1 Million+

Questions answered

;