QQuestion
In the given diagram, ABC is a triangle, where B(4, −4) and C(−4, −2). D is a point on AC.
(a) Write down the coordinates of A and D.
(b) Find the coordinates of the centroid of △ABC.
(c) If D divides AC in the ratio k : 1, find the value of k.
(d) Find the equation of the line BD.
✓Answer
(a) From the graph:
A = (0, 6) and D = (−3, 0).
(b) The centroid of a triangle with vertices (x₁, y₁), (x₂, y₂) and (x₃, y₃) is:
\left(\frac{x_1+x_2+x_3}{3},\frac{y_1+y_2+y_3}{3}\right)For A(0, 6), B(4, −4) and C(−4, −2):
Centroid = \left(\frac{0+4-4}{3},\frac{6-4-2}{3}\right)
⇒ Centroid = (0, 0)
(c) D divides AC in the ratio k : 1.
Using the section formula:
D=\left(\frac{k(-4)+1(0)}{k+1},\frac{k(-2)+1(6)}{k+1}\right)Since D = (−3, 0):
-3=\frac{-4k}{k+1}⇒ −3(k + 1) = −4k
⇒ −3k − 3 = −4k
⇒ k = 3
Checking the y-coordinate:
0=\frac{-2k+6}{k+1}⇒ −2k + 6 = 0
⇒ k = 3
(d) The line BD passes through B(4, −4) and D(−3, 0).
Slope = \frac{0-(-4)}{-3-4}=\frac{4}{-7}=-\frac{4}{7}
Using the point-slope form with B(4, −4):
y − (−4) = -\frac{4}{7}(x − 4)
⇒ 7(y + 4) = −4(x − 4)
⇒ 7y + 28 = −4x + 16
⇒ 4x + 7y + 12 = 0
Therefore, the equation of BD is 4x + 7y + 12 = 0.
Related Questions
More ICSE Hard level questions
A and B are two points on the x-axis and y-axis respectively, as shown in the figure. (a) Write down the coordinates of A and B. (b) P is a point on AB such that AP : PB = 3 : 1. Using the section formula, find the coordinates of P. (c) Find the equation of a line passing through P and perpendicular to AB.
ABC is a triangle whose vertices are A(1, −1), B(0, 4) and C(−6, 4). D is the midpoint of BC. Find: (a) the coordinates of D (b) the equation of the median AD.
The coordinates of the vertices of triangle ABC are respectively (−4, −2), (6, 2) and (4, 6). The centroid G of triangle ABC is: (a) (2, 2) (b) (2, 3) (c) (3, 3) (d) (0, −1)
In the given diagram, △ADB and △ACB are two right-angled triangles with ∠ADB = ∠BCA = 90°. If AB = 10 cm, AD = 6 cm, BC = 2.4 cm and DP = 4.5 cm: (a) Prove that △APD ∼ △BPC. (b) Find the lengths of BD and PB. (c) Hence, find the length of PA. (d) Find area of △APD : area of △BPC.
Use ruler and compass to answer this question. Construct ∠ABC = 90°, where AB = 6 cm and BC = 8 cm. (a) Construct the locus of points equidistant from B and C. (b) Construct the locus of points equidistant from A and B. (c) Mark the point which satisfies both conditions (a) and (b) as O. Construct the locus of points keeping a fixed distance OA from the fixed point O. (d) Construct the locus of points which are equidistant from BA and BC.
In the given diagram, O is the centre of the circle. PR and PT are two tangents drawn from the external point P and touching the circle at Q and S respectively. MN is a diameter of the circle. Given ∠PQM = 42° and ∠PSM = 25°. Find: (a) ∠OQM (b) ∠QNS (c) ∠QOS (d) ∠QMS
Prove that:
Mrs Rao deposited ₹250 per month in a recurring deposit account for 3 years. She received ₹10,110 at maturity. Find: (a) the rate of interest. (b) how much more interest she would receive if she deposited ₹50 more per month at the same rate and for the same time.
Mr and Mrs Das travelled by car from Delhi to Kasauli, a distance of approximately 350 km. Heavy rain reduced the average speed by 20 km/h and increased the journey time by 2 hours. Find:(a) the original speed of the car.(b) the time taken at the reduced speed.
A solid metallic cylinder is cut into two identical halves along its height. The diameter of the cylinder is 7 cm and the height is 10 cm. Find: (a) The total surface area of both the halves. (b) The total cost of painting the two halves at the rate of ₹30 per cm². (Use π = 22/7)
A life insurance agent found the following distribution of ages of 100 policy holders: Age (years) Frequency Cumulative frequency 20–25 2 2 25–30 4 6 30–35 12 18 35–40 20 38 40–45 28 66 45–50 22 88 50–55 8 96 55–60 4 100 On a graph sheet, draw an ogive using the given data. Take 2 cm = 5 years along one axis and 2 cm = 10 policy holders along the other axis. Use your graph to find: (a) the median age. (b) the number of policy holders whose age is above 52 years.
Oil is stored in a spherical vessel, occupying of its full capacity. The radius of the spherical vessel is 28 cm. This oil is then poured into a cylindrical vessel with radius 21 cm. Find the height of the oil in the cylindrical vessel, correct to the nearest centimetre. (Take π = )