Centroid, orthocentre and circumcentre of a triangle — practice questions

Question 1 · difficulty L3 · understanding

In an equilateral triangle $P Q R$, let the vertex $P$ be at $(3,5)$ and the side $Q R$ be along the line $x+y=4$. If the orthocentre of the triangle PQR is $(\alpha, \beta)$, then $9(\alpha+\beta)$ is equal to:

• A. 16
• B. 27
• C. 36
• D. 48

Question 2 · difficulty L3 · understanding

Let the vertex A of a triangle ABC be $(1,2)$, and the mid-point of the side AB be $(5,-1)$. If the centroid of this triangle is $(3,4)$ and its circumcenter is $(\alpha, \beta)$, then $21(\alpha+\beta)$ is equal to :

• A. 309
• B. 403
• C. 497
• D. 524

Question 3 · difficulty L3 · understanding

Let the mid points of the sides of a triangle ABC be $\left(\frac{5}{2}, 7\right)$, $\left(\frac{5}{2}, 3\right)$ and $(4, 5)$. If its incentre is $(h, k)$, then $3h + k$ is equal to :

• A. 11
• B. 12
• C. 13
• D. 14

Question 4 · difficulty L3 · understanding

If the orthocenter of the triangle formed by the lines y = x + 1, y = 4x - 8 and y = mx + c is at (3, -1), then m - c is :

• A. 0
• B. 2
• C. -2
• D. 4

Question 5 · difficulty L3 · understanding

Let ABC be the triangle such that the equations of lines AB and AC be $3 y-x=2$ and $x+y=2$, respectively, and the points B and C lie on $x$-axis. If P is the orthocentre of the triangle ABC , then the area of the triangle PBC is equal to

• A. 8
• B. 4
• C. 10
• D. 6

Question 6 · difficulty L3 · understanding

Let the three sides of a triangle are on the lines $4 x-7 y+10=0, x+y=5$ and $7 x+4 y=15$. Then the distance of its orthocentre from the orthocentre of the tringle formed by the lines $x=0, y=0$ and $x+y=1$ is

• A. $\sqrt{20}$
• B. $20$
• C. $\sqrt{5}$
• D. $5$

Question 7 · difficulty L3 · understanding

Let ΔABC be a triangle formed by the lines 7x – 6y + 3 = 0, x + 2y – 31 = 0 and 9x – 2y – 19 = 0. Let the point (h, k) be the image of the centroid of ΔABC in the line 3x + 6y – 53 = 0. Then h 2 + k 2 + hk is equal to :

• A. 47
• B. 37
• C. 40
• D. 36

Question 8 · difficulty L3 · understanding

Let the triangle PQR be the image of the triangle with vertices $(1,3),(3,1)$ and $(2,4)$ in the line $x+2 y=2$. If the centroid of $\triangle \mathrm{PQR}$ is the point $(\alpha, \beta)$, then $15(\alpha-\beta)$ is equal to :

• A. 21
• B. 19
• C. 22
• D. 24

Question 9 · difficulty L3 · understanding

If $(\alpha, \beta)$ is the orthocenter of the triangle $\mathrm{ABC}$ with vertices $A(3,-7), B(-1,2)$ and $C(4,5)$, then $9 \alpha-6 \beta+60$ is equal to :

• A. 30
• B. 40
• C. 25
• D. 35

Question 10 · difficulty L3 · understanding

Let $$(\alpha, \beta)$$ be the centroid of the triangle formed by the lines $$15 x-y=82,6 x-5 y=-4$$ and $$9 x+4 y=17$$. Then $$\alpha+2 \beta$$ and $$2 \alpha-\beta$$ are the roots of the equation :

• A. $$x^{2}-7 x+12=0$$
• B. $$x^{2}-13 x+42=0$$
• C. $$x^{2}-14 x+48=0$$
• D. $$x^{2}-10 x+25=0$$