Special centres and lines in a Triangle

Let O be the origin. If the coordinates of the points P and Q are (-24, 0) and (-24, 50) respectively, then the x-coordinate of the circumcentre of ΔOPQ is

• -24
• -12
• -8
• 25

If ΔABC is an acute-angled triangle, which of the following points must lie inside ΔABC?
I. The incentre of ΔABC
II. The circumcentre of ΔABC
III. The orthocentre of ΔABC

• I and II only
• I and III only
• II and III only
• I, II and III

In the figure, ΔPQR is an acute-angled triangle. S is a point on QR such that PS⊥QR and ∠QPS = ∠RPS. Which of the following are true?
I. The incentre of ΔPQR lies on PS.
II. The orthocentre of ΔPQR lies on PS.
III. The circumcentre of ΔPQR lies on PS.

• I and II only
• I and III only
• II and III only
• I, II and III

In ΔABC, ∠A : ∠B : ∠C = 5 : 18 : 13 and O is the circumcentre of ΔABC. Which of the following is / are true?
I. OA = OB = OC
II. O lies outside ΔABC
III. The orthocentre of ΔABC lies at the mid-point of AC.

• I only
• III only
• I and II only
• II and III only

In ΔABC, AB = AC and ∠A = 90°. Which of the following is / are true?
I. The circumcentre and the orthocentre of ΔABC coincide.
II. The incentre of ΔABC does not lie on the perpendicular bisector of BC.
III. The circumcentre, centroid, incentre and orthocentre of ΔABC are collinear.

• I only
• III only
• I and II only
• II and III only

In the figure, PX, QY and RZ intersect at H, where H is the centroid of ΔPQR. If PX + QY + RZ = 24 cm, find PH + QH + RH.

• 8 cm
• 12 cm
• 16 cm
• 36 cm

In the figure, G is the centroid of ΔABC, AGP and CGQ are straight lines. The ratio of the area of ΔCPG to the area of BPGQ is

• 1 : 1
• 1 : 2
• 2 : 1
• 3 : 2

In the figure, T is the incentre of ΔXYZ. If ∠YTZ = 180° − ∠TXZ, which of the following must be true?
I. ΔXYZ is a right-angled triangle.
II. ∠XYZ = 45°
III. ΔYTZ is an isosceles triangle.

• I only
• II only
• I and III only
• II and III only