Construct the circumcentre of the triangle XYZ shown below, using only a compass and straight edge. Label the circumcentre C. Show your construction lines clearly. The points A, B, C, and D lie on a circle, as shown in the diagram on the right (not to scale). [AB] is a diameter of the circle. |DAC| = 40°, as shown. The triangle ABD is isosceles. Find |ADC|. The diagram on the right shows the triangle PQR (not to scale). The angle at R is greater than 90°. k is the circumcircle of PQR, as shown, and O is the circumcentre. O is not shown in the diagram. Prove that O cannot be inside the triangle PQR. If you are proving this by contradiction, your first line should be: "Assume that O is inside the triangle PQR."

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Construct the circumcentre of the triangle XYZ shown below, using only a compass and straight edge - Leaving Cert Mathematics - Question 6 - 2022

Question 6

Construct the circumcentre of the triangle XYZ shown below, using only a compass and straight edge. Label the circumcentre C. Show your construction lines clearly. ... show full transcript

Worked Solution & Example Answer:Construct the circumcentre of the triangle XYZ shown below, using only a compass and straight edge - Leaving Cert Mathematics - Question 6 - 2022

Step 1

Construct the circumcentre of the triangle XYZ

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Answer

Begin by drawing the triangle XYZ on your paper.
Use the compass to construct the perpendicular bisector of segment XY.
- To do this, place the compass point on point X and draw an arc above and below the line.
- Without changing the compass width, place the compass point on point Y and draw two arcs, intersecting the first two arcs.
Label the intersection points of the arcs as points P and Q.
Draw a straight line through points P and Q. This will be the perpendicular bisector of XY.
Repeat steps 2 to 4 for segment XZ to find its perpendicular bisector, and label its intersection points as R and S.
The point where the two perpendicular bisectors intersect is the circumcentre C of triangle XYZ.

Step 2

Find |ADC|.

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Answer

To find |ADC|:

Note that since [AB] is the diameter, |ADB| = 90° (angles in a semi-circle).
Since triangle ADB is isosceles, we have: |BAD| = |ABD|.
Given that |DAC| = 40°, we can set up the equation: |ADC| = 90° - |BAD|.
Since |BAD| and |ABD| are equal, we can denote |BAD| = x.
The equation becomes: $x + 40° + x = 90°$ .
Simplifying gives: $2x + 40° = 90°$ .
Thus, $2x = 50°$ → $x = 25°$ .
Therefore, |ADC| = 90° - 25° = 65°.

Step 3

Prove that O cannot be inside the triangle PQR.

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Answer

Assume that O is inside the triangle PQR.

The angle at the centre I, ∠[P O Q], must be twice the angle at point R: |PQR|, hence:

∠[POQ] must be > 180°.
However, since we know that the angle at R is greater than 90°, it follows that: |PQR| must also be greater than 90°.
This leads to a contradiction, as the circumcentre O cannot exist within the triangle when the angle exceeds 90°.
Therefore, it must be concluded that O cannot be inside triangle PQR.

Construct the circumcentre of the triangle XYZ shown below, using only a compass and straight edge - Leaving Cert Mathematics - Question 6 - 2022

Worked Solution & Example Answer:Construct the circumcentre of the triangle XYZ shown below, using only a compass and straight edge - Leaving Cert Mathematics - Question 6 - 2022

Construct the circumcentre of the triangle XYZ

Find |ADC|.

Prove that O cannot be inside the triangle PQR.

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