Junior Secondary · Mathematics

Centers of Triangles: Practice Questions

5 multiple-choice questions marked as you go, and 5 written questions with worked solutions. All on Centers of Triangles.

10 questions23 marksFree, no account

Question 1

1 mark

In \(\triangle PQR\), if the point O is the circumcentre of the triangle and the distance from O to vertex P is \(15\) cm, what is the distance from point O to vertex R?

Question 2

1 mark

The area of \(\triangle ABC\) is \(54\text{ cm}^2\). If the incenter \(I\) of the triangle is at a perpendicular distance of \(3\text{ cm}\) from the side \(AB\), calculate the perimeter of \(\triangle ABC\).

Question 3

1 mark

In \(\triangle ABC\), the medians \(AD\) and \(BE\) intersect at the centroid \(G\). If \(AD = 18\) cm, \(BE = 15\) cm, and the medians are perpendicular to each other, find the area of \(\triangle ABC\).

Question 4

1 mark

Point P lies on the perpendicular bisector of the line segment AB. Which of the following relationships must be true?

Question 5

1 mark

In \(\triangle ABC\), \(G\) is the centroid. Points \(D\) and \(F\) are the midpoints of sides \(BC\) and \(AB\) respectively. If the area of \(\triangle ABC\) is \(54 \text{ cm}^2\), what is the area of the quadrilateral \(BDFG\)?

Question 6

2 marks

If the three angle bisectors of \(\triangle PQR\) intersect at point \(I\), and the perpendicular distance from \(I\) to side \(PQ\) is \(7\) mm, what is the perpendicular distance from \(I\) to side \(QR\)?

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Question 7

3 marks

Point \(O\) is the circumcenter of \(\triangle PQR\). If the distance from \(O\) to the vertex \(P\) is \(9\) cm, find the diameter of the circumcircle of the triangle.

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Question 8

5 marks

In \(\triangle ABC\), the incenter \(I\) is located such that the distance from \(I\) to the side \(BC\) is \(3\) cm. If the perimeter of the triangle is \(30\) cm, calculate the area of \(\triangle ABC\).

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Question 9

3 marks

In $$\triangle LMN$$ , the perpendicular bisectors of the sides intersect at point $$O$$ . This point $$O$$ is the circumcentre of the triangle.

(a) If the length of $$OL$$ is $$15\text{ cm}$$ , what is the length of $$OM$$ and $$ON$$ ?
(b) State the definition of the circumcentre in terms of the vertices of the triangle.

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Question 10

5 marks

In \(\triangle PQR\), the point \(M\) is the midpoint of side \(QR\). Point \(G\) is the centroid of the triangle and the median \(PM\) has a total length of \(27 \text{ cm}\).

(a) Find the lengths of the segments \(PG\) and \(GM\).
(b) The area of \(\triangle PGM\) is \(10 \text{ cm}^2\). By considering the ratio of the bases \(PM\) and \(GM\) for triangles sharing the same altitude from vertex \(Q\), find the area of \(\triangle PQM\). Explain your reasoning.
(c) Determine the total area of \(\triangle PQR\) using the property that a median divides a triangle into two regions of equal area.

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