The diagram shows a semi-circle standing on a diameter [AC], and [BD] ⊥ [AC]. (a) (i) Prove that the triangles ABD and DBC are similar. (ii) If |AB| = x, |BC| = 1, and |BD| = y, write y in terms of x. (b) Use your result from part (a)(ii) to construct a line segment equal in length (in centimetres) to the square root of the length of the line segment [TU] which is drawn below.

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The diagram shows a semi-circle standing on a diameter [AC], and [BD] ⊥ [AC] - Leaving Cert Mathematics - Question 4 - 2016

Question 4

The diagram shows a semi-circle standing on a diameter [AC], and [BD] ⊥ [AC]. (a) (i) Prove that the triangles ABD and DBC are similar. (ii) If |AB| = x, |BC| = 1,... show full transcript

Worked Solution & Example Answer:The diagram shows a semi-circle standing on a diameter [AC], and [BD] ⊥ [AC] - Leaving Cert Mathematics - Question 4 - 2016

Step 1

Prove that the triangles ABD and DBC are similar.

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Answer

To show that triangles ABD and DBC are similar, we can use the Angle-Angle (AA) similarity criterion.

Consider triangle ABD:
- Angle |ABD| = 90° (angle in a semicircle)
- Angle |ABD| = |ACD| = |DCA|
Now consider triangle DBC:
- Angle |DBC| = 90° (since [BD] is perpendicular to [AC])
- Angle |BCD| = |ABD| (angles subtended by the same arc)

Since both triangles share the angle |BDC|, we know:

|ABD| = 90°
|ABC| = 90° Thus, we can conclude that:

$\text{Triangles } ABD \text{ and } DBC \text{ are similar.}$

Step 2

If |AB| = x, |BC| = 1, and |BD| = y, write y in terms of x.

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Answer

To find y in terms of x, we can apply the Pythagorean theorem in triangle ABD.

We have:

|AB| = x
|BC| = 1
|BD| = y

Using the Pythagorean theorem: $|AD|^2 = |AB|^2 + |BD|^2 \\ |AC|^2 = x^2 + y^2$

Also, from triangle DBC: $|BD|^2 + |BC|^2 = |DC|^2 \\ y^2 + 1^2 = |DC|^2 \\$ Let |DC| = \sqrt{x^2 + y^2} \\ y^2 + 1 = x^2 + y^2$

So we now have two equations that can be equalized and solved for y: $2y^2 = x^2 - 1 \\$ \\ y = \sqrt{x^2 - 1}$

Step 3

Use your result from part (a)(ii) to construct a line segment equal in length (in centimetres) to the square root of the length of the line segment [TU].

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Answer

To construct the segment:

Measure the length of segment [TU].
Calculate the square root of the measured length using the previously derived relationships.
Using a compass, construct a segment of length equal to the square root of [TU]. This will create the required segment that matches the length you've calculated. Ensure that this segment is drawn accurately for precision.

The diagram shows a semi-circle standing on a diameter [AC], and [BD] ⊥ [AC] - Leaving Cert Mathematics - Question 4 - 2016

Worked Solution & Example Answer:The diagram shows a semi-circle standing on a diameter [AC], and [BD] ⊥ [AC] - Leaving Cert Mathematics - Question 4 - 2016

Prove that the triangles ABD and DBC are similar.

If |AB| = x, |BC| = 1, and |BD| = y, write y in terms of x.

Use your result from part (a)(ii) to construct a line segment equal in length (in centimetres) to the square root of the length of the line segment [TU].

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