Centers of Triangles：練習問題

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?

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 \(\triangle ABC\).

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 \(\triangle ABC\).

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

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 \(BDFG\)?

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 \(QR\)?

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.

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 \(\triangle ABC\).

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

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.