ONE-EIGHTH STAGE 2025
Anlo SHS: 31 points
Mpraeso SHS: 35 points
Mfantsiman Girls' SHS: 27 points
Pope John SHS & Minor Seminary: 58 points
Ghana SHS, Tamale: 25 points
Awudome SHS: 23 points
Swedru Secondary School: 36 points
Kumasi High School: 31 points
Anlo SHS: 30 points
ROUND 4 - TRUE OR FALSE
PREAMBLE
For any triangle the given point lies inside the triangle.
1. Circumcenter
ANSWER: FALSE
EXPLANATION:
The circumcenter is the intersection of the perpendicular bisectors.
It lies inside an acute triangle.
It lies on the hypotenuse (its midpoint) of a right triangle.
It lies outside an obtuse triangle.
So it does not always lie inside the triangle.
2. Centroid
ANSWER: TRUE
EXPLANATION:
The centroid is the intersection of the medians.
It lies inside every triangle (acute, right or obtuse).
3. Orthocenter
ANSWER: FALSE
EXPLANATION:
The orthocenter is the intersection of the altitudes.
It lies inside an acute triangle.
It is at the right-angle vertex of a right triangle.
It lies outside an obtuse triangle.
So it does not always lie inside the triangle.
NOTES ON TRIANGLE CENTERS
1.
Centroid: the intersection of the three medians.
It divides each median in the ratio $2:1$ from the vertex.
It is the balancing point of the triangle and always lies inside.
2.
Circumcenter: the intersection of the perpendicular bisectors of the sides.
It is the centre of the circle through all three vertices, and it is equidistant from the vertices.
3.
Orthocenter: the intersection of the three altitudes.
4.
Incenter: the intersection of the angle bisectors.
It is the centre of the circle tangent to all three sides, and it always lies inside the triangle.
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PRACTICE QUESTIONS
PREAMBLE 1
The given point lies inside every acute triangle:
1. Orthocentre
ANSWER: TRUE
EXPLANATION:
In an acute triangle, all three altitudes meet inside.
2. Circumcentre
ANSWER: TRUE
EXPLANATION:
It lies inside an acute triangle.
3. A vertex
ANSWER: FALSE
EXPLANATION:
A vertex lies on the triangle, not inside it.
PREAMBLE 2
For a right-angled triangle:
1. The orthocentre is at the right-angle vertex.
ANSWER: TRUE
EXPLANATION:
The two shorter sides are altitudes.
2. The circumcentre is the midpoint of the hypotenuse.
ANSWER: TRUE
EXPLANATION:
The hypotenuse is a diameter of the circumcircle.
3. The incentre lies on the hypotenuse.
ANSWER: FALSE
EXPLANATION:
The incentre always lies inside the triangle.
PREAMBLE 3
For an obtuse triangle:
1. The circumcentre lies outside the triangle.
ANSWER: TRUE
EXPLANATION:
It lies beyond the longest side.
2. The orthocentre lies outside the triangle.
ANSWER: TRUE
EXPLANATION:
Two of the altitudes fall outside.
3. The centroid lies outside the triangle.
ANSWER: FALSE
EXPLANATION:
The centroid is always inside.