CAT Exam > CAT Questions > If C is the centroid of the triangle PQR and ...
Start Learning for Free
If C is the centroid of the triangle PQR and the coordinates of points P, Q, R and C respectively are (x, 1), (0, y), (1/2, 0) and (1/2, 1/3), find the measure of angle PQR.
- a)55°
- b)50°
- c)53°
- d)45°
Correct answer is option 'D'. Can you explain this answer?
Most Upvoted Answer
If C is the centroid of the triangle PQR and the coordinates of points...
The centroid of ΔPQR has coordinates (1/2, 1/3).
Therefore, 1/2 = (x + 0 + 1/2)/3 and 1/3 = (1 + y + 0)/3
So, x = 1 and y = 0
Angle PQR is equal to angle between line PQ and line QR.
Slope of PQ (m1)= (0 - 1)/(0 - 1) = 1
Slope of QR (m2)= (0 - 0)/((1/2) - 0) = 0
Angle between line PQ and line QR = tan θ = (m1 - m2)/(1 + m1m2) = (1 - 0)/(1 + (0)(1)) = 1
tan θ = 1
θ = 45°
Therefore, 1/2 = (x + 0 + 1/2)/3 and 1/3 = (1 + y + 0)/3
So, x = 1 and y = 0
Angle PQR is equal to angle between line PQ and line QR.
Slope of PQ (m1)= (0 - 1)/(0 - 1) = 1
Slope of QR (m2)= (0 - 0)/((1/2) - 0) = 0
Angle between line PQ and line QR = tan θ = (m1 - m2)/(1 + m1m2) = (1 - 0)/(1 + (0)(1)) = 1
tan θ = 1
θ = 45°
Free Test
FREE
| Start Free Test |
Community Answer
If C is the centroid of the triangle PQR and the coordinates of points...
To find the measure of angle PQR, we need to find the coordinates of points P, Q, and R.
Since C is the centroid of triangle PQR, the coordinates of C are the average of the coordinates of P, Q, and R. Thus, we can find the coordinates of P, Q, and R by doubling the coordinates of C and subtracting the coordinates of Q and R respectively.
Coordinates of P = 2C - Q = 2(1/2, 1/3) - (0, y) = (1, 2/3) - (0, y) = (1, 2/3 - y)
Coordinates of R = 2C - R = 2(1/2, 1/3) - (1/2, 0) = (1, 2/3) - (1/2, 0) = (1 - 1/2, 2/3 - 0) = (1/2, 2/3)
Let's find the value of y by equating the y-coordinates of P and R:
2/3 - y = 2/3
y = 0
So, the coordinates of P are (1, 2/3 - 0) = (1, 2/3) and the coordinates of R are (1/2, 2/3).
Now, let's find the measure of angle PQR. We can use the Law of Cosines to find the angle.
Let PQ = a, QR = b, and RP = c.
Using the distance formula, we can find the lengths of sides PQ, QR, and RP:
PQ = sqrt((1 - 0)^2 + (2/3 - 0)^2) = sqrt(1 + 4/9) = sqrt(13/9)
QR = sqrt((1/2 - 0)^2 + (2/3 - 1)^2) = sqrt(1/4 + 1/9) = sqrt(13/36)
RP = sqrt((1 - 1/2)^2 + (2/3 - 2/3)^2) = sqrt(1/4) = 1/2
Now, let's use the Law of Cosines to find the measure of angle PQR:
cos PQR = (PQ^2 + QR^2 - RP^2) / (2 * PQ * QR)
cos PQR = (13/9 + 13/36 - (1/2)^2) / (2 * sqrt(13/9) * sqrt(13/36))
cos PQR = (13/9 + 13/36 - 1/4) / (2 * sqrt(13/9) * sqrt(13/36))
cos PQR = (52/36 + 13/36 - 9/36) / (2 * sqrt(13/9) * sqrt(13/36))
cos PQR = (56/36) / (2 * sqrt(13/9) * sqrt(13/36))
cos PQR = (7/9) / (2 * sqrt(13/9) * sqrt(13/36))
cos PQR = (7/9) / (2 * (sqrt(13)/3) * (sqrt(
Since C is the centroid of triangle PQR, the coordinates of C are the average of the coordinates of P, Q, and R. Thus, we can find the coordinates of P, Q, and R by doubling the coordinates of C and subtracting the coordinates of Q and R respectively.
Coordinates of P = 2C - Q = 2(1/2, 1/3) - (0, y) = (1, 2/3) - (0, y) = (1, 2/3 - y)
Coordinates of R = 2C - R = 2(1/2, 1/3) - (1/2, 0) = (1, 2/3) - (1/2, 0) = (1 - 1/2, 2/3 - 0) = (1/2, 2/3)
Let's find the value of y by equating the y-coordinates of P and R:
2/3 - y = 2/3
y = 0
So, the coordinates of P are (1, 2/3 - 0) = (1, 2/3) and the coordinates of R are (1/2, 2/3).
Now, let's find the measure of angle PQR. We can use the Law of Cosines to find the angle.
Let PQ = a, QR = b, and RP = c.
Using the distance formula, we can find the lengths of sides PQ, QR, and RP:
PQ = sqrt((1 - 0)^2 + (2/3 - 0)^2) = sqrt(1 + 4/9) = sqrt(13/9)
QR = sqrt((1/2 - 0)^2 + (2/3 - 1)^2) = sqrt(1/4 + 1/9) = sqrt(13/36)
RP = sqrt((1 - 1/2)^2 + (2/3 - 2/3)^2) = sqrt(1/4) = 1/2
Now, let's use the Law of Cosines to find the measure of angle PQR:
cos PQR = (PQ^2 + QR^2 - RP^2) / (2 * PQ * QR)
cos PQR = (13/9 + 13/36 - (1/2)^2) / (2 * sqrt(13/9) * sqrt(13/36))
cos PQR = (13/9 + 13/36 - 1/4) / (2 * sqrt(13/9) * sqrt(13/36))
cos PQR = (52/36 + 13/36 - 9/36) / (2 * sqrt(13/9) * sqrt(13/36))
cos PQR = (56/36) / (2 * sqrt(13/9) * sqrt(13/36))
cos PQR = (7/9) / (2 * sqrt(13/9) * sqrt(13/36))
cos PQR = (7/9) / (2 * (sqrt(13)/3) * (sqrt(
|
Explore Courses for CAT exam
|
|
Similar CAT Doubts
Top Courses for CATView all
If C is the centroid of the triangle PQR and the coordinates of points P, Q, R and C respectively are (x, 1), (0, y), (1/2, 0) and (1/2, 1/3), find the measure of angle PQR.a)55°b)50°c)53°d)45°Correct answer is option 'D'. Can you explain this answer?
Question Description
If C is the centroid of the triangle PQR and the coordinates of points P, Q, R and C respectively are (x, 1), (0, y), (1/2, 0) and (1/2, 1/3), find the measure of angle PQR.a)55°b)50°c)53°d)45°Correct answer is option 'D'. Can you explain this answer? for CAT 2025 is part of CAT preparation. The Question and answers have been prepared according to the CAT exam syllabus. Information about If C is the centroid of the triangle PQR and the coordinates of points P, Q, R and C respectively are (x, 1), (0, y), (1/2, 0) and (1/2, 1/3), find the measure of angle PQR.a)55°b)50°c)53°d)45°Correct answer is option 'D'. Can you explain this answer? covers all topics & solutions for CAT 2025 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for If C is the centroid of the triangle PQR and the coordinates of points P, Q, R and C respectively are (x, 1), (0, y), (1/2, 0) and (1/2, 1/3), find the measure of angle PQR.a)55°b)50°c)53°d)45°Correct answer is option 'D'. Can you explain this answer?.
If C is the centroid of the triangle PQR and the coordinates of points P, Q, R and C respectively are (x, 1), (0, y), (1/2, 0) and (1/2, 1/3), find the measure of angle PQR.a)55°b)50°c)53°d)45°Correct answer is option 'D'. Can you explain this answer? for CAT 2025 is part of CAT preparation. The Question and answers have been prepared according to the CAT exam syllabus. Information about If C is the centroid of the triangle PQR and the coordinates of points P, Q, R and C respectively are (x, 1), (0, y), (1/2, 0) and (1/2, 1/3), find the measure of angle PQR.a)55°b)50°c)53°d)45°Correct answer is option 'D'. Can you explain this answer? covers all topics & solutions for CAT 2025 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for If C is the centroid of the triangle PQR and the coordinates of points P, Q, R and C respectively are (x, 1), (0, y), (1/2, 0) and (1/2, 1/3), find the measure of angle PQR.a)55°b)50°c)53°d)45°Correct answer is option 'D'. Can you explain this answer?.
Solutions for If C is the centroid of the triangle PQR and the coordinates of points P, Q, R and C respectively are (x, 1), (0, y), (1/2, 0) and (1/2, 1/3), find the measure of angle PQR.a)55°b)50°c)53°d)45°Correct answer is option 'D'. Can you explain this answer? in English & in Hindi are available as part of our courses for CAT.
Download more important topics, notes, lectures and mock test series for CAT Exam by signing up for free.
Here you can find the meaning of If C is the centroid of the triangle PQR and the coordinates of points P, Q, R and C respectively are (x, 1), (0, y), (1/2, 0) and (1/2, 1/3), find the measure of angle PQR.a)55°b)50°c)53°d)45°Correct answer is option 'D'. Can you explain this answer? defined & explained in the simplest way possible. Besides giving the explanation of
If C is the centroid of the triangle PQR and the coordinates of points P, Q, R and C respectively are (x, 1), (0, y), (1/2, 0) and (1/2, 1/3), find the measure of angle PQR.a)55°b)50°c)53°d)45°Correct answer is option 'D'. Can you explain this answer?, a detailed solution for If C is the centroid of the triangle PQR and the coordinates of points P, Q, R and C respectively are (x, 1), (0, y), (1/2, 0) and (1/2, 1/3), find the measure of angle PQR.a)55°b)50°c)53°d)45°Correct answer is option 'D'. Can you explain this answer? has been provided alongside types of If C is the centroid of the triangle PQR and the coordinates of points P, Q, R and C respectively are (x, 1), (0, y), (1/2, 0) and (1/2, 1/3), find the measure of angle PQR.a)55°b)50°c)53°d)45°Correct answer is option 'D'. Can you explain this answer? theory, EduRev gives you an
ample number of questions to practice If C is the centroid of the triangle PQR and the coordinates of points P, Q, R and C respectively are (x, 1), (0, y), (1/2, 0) and (1/2, 1/3), find the measure of angle PQR.a)55°b)50°c)53°d)45°Correct answer is option 'D'. Can you explain this answer? tests, examples and also practice CAT tests.
|
|
Explore Courses for CAT exam
|
|
Signup for Free!
Signup to see your scores go up within 7 days! Learn & Practice with 1000+ FREE Notes, Videos & Tests.
|
© EduRev
|
Education Revolution
|
|
Signup to see your scores
go up within 7 days!
Access 1000+ FREE Docs, Videos and Tests
Takes less than 10 seconds to signup