Important Formulas
Inclination of a line:
0° ≤ θ < 180°
Slope of a non-vertical line:
m = tan θ
Slope using 2 points:
If the points are (x1, y1) and (x2, y2), then
m = (y2 - y1) / (x2 - x1)
Slope of a horizontal line:
m = 0
Slope of a vertical line:
m is not defined
Condition for parallel lines:
If L1 ∥ L2, then m1 = m2
Condition for perpendicular lines:
If L1 ⟂ L2, then m1m2 = -1
Angle between 2 non-vertical and non-perpendicular lines:
tan α = |(m2 - m1) / (1 + m1m2)|
Example 1
Question in English:
Find the slope of a line whose inclination is:
(i) 45°
(ii) 60°
(iii) 150°
Question in Hindi:
उस रेखा की ढाल ज्ञात कीजिए जिसका झुकाव है:
(i) 45°
(ii) 60°
(iii) 150°
Solution:
We know that:
m = tan θ
(i) When θ = 45°,
m = tan 45° = 1
(ii) When θ = 60°,
m = tan 60° = √3
(iii) When θ = 150°,
m = tan 150°
m = tan (180° - 30°)
m = -tan 30°
m = -1 / √3
Answer:
(i) 1
(ii) √3
(iii) -1 / √3
Example 2
Question in English:
What is the inclination of a line whose slope is:
(i) 0
(ii) positive
(iii) negative
(iv) not defined?
Question in Hindi:
उस रेखा का झुकाव क्या होगा जिसकी ढाल है:
(i) 0
(ii) धनात्मक
(iii) ऋणात्मक
(iv) अपरिभाषित?
Solution:
Let θ be the inclination of the given line.
We know that:
m = tan θ
Also,
0° ≤ θ < 180°
(i) When m = 0,
tan θ = 0
Therefore,
θ = 0°
(ii) When m > 0,
tan θ > 0
Therefore, θ lies between 0° and 90°.
Hence, θ is acute.
(iii) When m < 0,
tan θ < 0
Therefore, θ lies between 90° and 180°.
Hence, θ is obtuse.
(iv) We know that a vertical line is the only line whose slope is not defined.
The inclination of a vertical line is 90°.
Hence, the inclination of a line whose slope is not defined is 90°.
Example 3
Question in English:
Find the slope of the line passing through the points:
(i) (-2, 3) and (8, -5)
(ii) (4, -3) and (6, -3)
(iii) (3, -1) and (3, 2)
Question in Hindi:
निम्नलिखित बिंदुओं से होकर गुजरने वाली रेखा की ढाल ज्ञात कीजिए:
(i) (-2, 3) और (8, -5)
(ii) (4, -3) और (6, -3)
(iii) (3, -1) और (3, 2)
Solution:
We know that:
m = (y2 - y1) / (x2 - x1)
(i) For the points (-2, 3) and (8, -5),
m = (-5 - 3) / (8 - (-2))
m = -8 / 10
m = -4 / 5
(ii) For the points (4, -3) and (6, -3),
m = (-3 - (-3)) / (6 - 4)
m = 0 / 2
m = 0
(iii) For the points (3, -1) and (3, 2),
m = (2 - (-1)) / (3 - 3)
m = 3 / 0
So, the slope is not defined.
Answer:
(i) -4 / 5
(ii) 0
(iii) Not defined
Example 4
Question in English:
If the slope of the line passing through the points (2, 5) and (x, 3) is 2, find the value of x.
Question in Hindi:
यदि (2, 5) और (x, 3) से गुजरने वाली रेखा की ढाल 2 है, तो x का मान ज्ञात कीजिए।
Solution:
We know that:
m = (y2 - y1) / (x2 - x1)
Therefore,
2 = (3 - 5) / (x - 2)
2 = -2 / (x - 2)
2(x - 2) = -2
2x - 4 = -2
2x = 2
x = 1
Answer:
x = 1
Example 5
Question in English:
Find the value of x so that the inclination of the line joining the points (x, -3) and (2, 5) is 135°.
Question in Hindi:
x का मान ज्ञात कीजिए ताकि (x, -3) और (2, 5) को मिलाने वाली रेखा का झुकाव 135° हो।
Solution:
Slope of the line:
m = (5 - (-3)) / (2 - x)
m = 8 / (2 - x)
We know that:
m = tan 135°
m = -1
Therefore,
8 / (2 - x) = -1
8 = x - 2
x = 10
Answer:
x = 10
Example 6
Question in English:
Find the angle between the x-axis and the line joining the points (3, -1) and (4, -2).
Question in Hindi:
x-अक्ष और (3, -1), (4, -2) को मिलाने वाली रेखा के बीच का कोण ज्ञात कीजिए।
Solution:
Slope of the line:
m = (-2 - (-1)) / (4 - 3)
m = -1 / 1
m = -1
Let the required angle be θ.
tan θ = -1
tan θ = tan 135°
Therefore,
θ = 135°
Answer:
135°
Example 7
Question in English:
Show that the line joining the points (2, -3) and (-5, 1) is parallel to the line joining the points (7, -1) and (0, 3).
Question in Hindi:
सिद्ध कीजिए कि (2, -3) और (-5, 1) को मिलाने वाली रेखा, (7, -1) और (0, 3) को मिलाने वाली रेखा के समांतर है।
Solution:
Slope of first line:
m1 = (1 - (-3)) / (-5 - 2)
m1 = 4 / -7
m1 = -4 / 7
Slope of second line:
m2 = (3 - (-1)) / (0 - 7)
m2 = 4 / -7
m2 = -4 / 7
Since,
m1 = m2
Therefore, the two lines are parallel.
Example 8
Question in English:
Show that the line joining the points (2, -5) and (-2, 5) is perpendicular to the line joining the points (6, 3) and (1, 1).
Question in Hindi:
सिद्ध कीजिए कि (2, -5) और (-2, 5) को मिलाने वाली रेखा, (6, 3) और (1, 1) को मिलाने वाली रेखा पर लंब है।
Solution:
Slope of first line:
m1 = (5 - (-5)) / (-2 - 2)
m1 = 10 / -4
m1 = -5 / 2
Slope of second line:
m2 = (1 - 3) / (1 - 6)
m2 = -2 / -5
m2 = 2 / 5
Now,
m1m2 = (-5 / 2) × (2 / 5)
m1m2 = -1
Hence, the lines are perpendicular.
Example 9
Question in English:
If the line through the points (-2, 6) and (4, 8) is perpendicular to the line through the points (8, 12) and (x, 24), find the value of x.
Question in Hindi:
यदि (-2, 6) और (4, 8) से गुजरने वाली रेखा, (8, 12) और (x, 24) से गुजरने वाली रेखा पर लंब है, तो x का मान ज्ञात कीजिए।
Solution:
Slope of first line:
m1 = (8 - 6) / (4 - (-2))
m1 = 2 / 6
m1 = 1 / 3
Slope of second line:
m2 = (24 - 12) / (x - 8)
m2 = 12 / (x - 8)
Since lines are perpendicular,
m1m2 = -1
(1 / 3) × (12 / (x - 8)) = -1
4 / (x - 8) = -1
4 = -(x - 8)
4 = -x + 8
x = 4
Answer:
x = 4
Example 10
Question in English:
Without using Pythagoras theorem, show that the points (4, 4), (3, 5), and (-1, -1) are the vertices of a right-angled triangle.
Question in Hindi:
पाइथागोरस प्रमेय का उपयोग किए बिना सिद्ध कीजिए कि (4, 4), (3, 5) और (-1, -1) एक समकोण त्रिभुज के शीर्ष हैं।
Solution:
Let the points be:
A(4, 4), B(3, 5), C(-1, -1)
Slope of AB:
m1 = (5 - 4) / (3 - 4)
m1 = 1 / -1
m1 = -1
Slope of AC:
m2 = (-1 - 4) / (-1 - 4)
m2 = -5 / -5
m2 = 1
Now,
m1m2 = (-1)(1)
m1m2 = -1
Therefore,
AB ⟂ AC
Hence,
∠CAB = 90°
Therefore, the given points form a right-angled triangle.
Example 11
Question in English:
Using slopes, show that the points (5, 1), (1, -1), and (11, 4) are collinear.
Question in Hindi:
ढालों का प्रयोग करके सिद्ध कीजिए कि बिंदु (5, 1), (1, -1), और (11, 4) संरेखीय हैं।
Solution:
Let the points be:
A(5, 1), B(1, -1), C(11, 4)
Slope of AB:
mAB = (-1 - 1) / (1 - 5)
mAB = -2 / -4
mAB = 1 / 2
Slope of BC:
mBC = (4 - (-1)) / (11 - 1)
mBC = 5 / 10
mBC = 1 / 2
Since,
mAB = mBC
Therefore, A, B, and C are collinear.
Example 12
Question in English:
Find the value of x for which the points (x, -1), (2, 1), and (4, 5) are collinear.
Question in Hindi:
x का मान ज्ञात कीजिए जिसके लिए बिंदु (x, -1), (2, 1), और (4, 5) संरेखीय हों।
Solution:
Let the points be:
A(x, -1), B(2, 1), C(4, 5)
Since the points are collinear,
Slope of AB = Slope of BC
(1 - (-1)) / (2 - x) = (5 - 1) / (4 - 2)
2 / (2 - x) = 4 / 2
2 / (2 - x) = 2
2 = 2(2 - x)
2 = 4 - 2x
2x = 2
x = 1
Answer:
x = 1
Example 13
Question in English:
If the points (h, 0), (a, b), and (0, k) lie on a line, show that:
a / h + b / k = 1
Question in Hindi:
यदि बिंदु (h, 0), (a, b), और (0, k) एक रेखा पर स्थित हैं, तो सिद्ध कीजिए:
a / h + b / k = 1
Solution:
Let the points be:
A(h, 0), B(a, b), C(0, k)
Since A, B, and C are collinear,
Slope of AB = Slope of BC
(b - 0) / (a - h) = (k - b) / (0 - a)
b / (a - h) = (k - b) / -a
-ab = (a - h)(k - b)
-ab = ak - ab - hk + hb
0 = ak - hk + hb
hk = ak + hb
Divide both sides by hk:
1 = a / h + b / k
Therefore,
a / h + b / k = 1
Hence proved.
Example 14
Question in English:
Using slopes, show that the vertices (-2, -1), (4, 0), (3, 3), and (-3, 2) are the vertices of a parallelogram.
Question in Hindi:
ढालों का प्रयोग करके सिद्ध कीजिए कि बिंदु (-2, -1), (4, 0), (3, 3), और (-3, 2) एक समांतर चतुर्भुज के शीर्ष हैं।
Solution:
Let the points be:
A(-2, -1), B(4, 0), C(3, 3), D(-3, 2)
Slope of AB:
mAB = (0 - (-1)) / (4 - (-2))
mAB = 1 / 6
Slope of DC:
mDC = (3 - 2) / (3 - (-3))
mDC = 1 / 6
Therefore,
mAB = mDC
So,
AB ∥ DC
Slope of BC:
mBC = (3 - 0) / (3 - 4)
mBC = 3 / -1
mBC = -3
Slope of AD:
mAD = (2 - (-1)) / (-3 - (-2))
mAD = 3 / -1
mAD = -3
Therefore,
mBC = mAD
So,
BC ∥ AD
Since both pairs of opposite sides are parallel, ABCD is a parallelogram.
Example 15
Question in English:
A(-4, 2), B(2, 6), C(8, 5), and D(9, -7) are the vertices of a quadrilateral ABCD. If P, Q, R, and S are the midpoints of AB, BC, CD, and DA respectively, using slopes, show that PQRS is a parallelogram.
Question in Hindi:
A(-4, 2), B(2, 6), C(8, 5), और D(9, -7) चतुर्भुज ABCD के शीर्ष हैं। यदि P, Q, R, और S क्रमशः AB, BC, CD, और DA के मध्यबिंदु हैं, तो ढालों का प्रयोग करके सिद्ध कीजिए कि PQRS एक समांतर चतुर्भुज है।
Solution:
Using midpoint formula:
P = ((-4 + 2) / 2, (2 + 6) / 2)
P = (-1, 4)
Q = ((2 + 8) / 2, (6 + 5) / 2)
Q = (5, 11 / 2)
R = ((8 + 9) / 2, (5 + (-7)) / 2)
R = (17 / 2, -1)
S = ((9 + (-4)) / 2, (-7 + 2) / 2)
S = (5 / 2, -5 / 2)
Slope of PQ:
mPQ = ((11 / 2) - 4) / (5 - (-1))
mPQ = (3 / 2) / 6
mPQ = 3 / 12
mPQ = 1 / 4
Slope of SR:
mSR = (-1 - (-5 / 2)) / ((17 / 2) - (5 / 2))
mSR = (3 / 2) / 6
mSR = 3 / 12
mSR = 1 / 4
Therefore,
mPQ = mSR
So,
PQ ∥ SR
Slope of QR:
mQR = (-1 - (11 / 2)) / ((17 / 2) - 5)
mQR = (-13 / 2) / (7 / 2)
mQR = -13 / 7
Slope of PS:
mPS = ((-5 / 2) - 4) / ((5 / 2) - (-1))
mPS = (-13 / 2) / (7 / 2)
mPS = -13 / 7
Therefore,
mQR = mPS
So,
QR ∥ PS
Since both pairs of opposite sides are parallel, PQRS is a parallelogram.
Example 16
Question in English:
Find the angle between the lines whose slopes are 1 / 2 and 3.
Question in Hindi:
उन रेखाओं के बीच का कोण ज्ञात कीजिए जिनकी ढालें 1 / 2 और 3 हैं।
Solution:
Let:
m1 = 1 / 2
m2 = 3
We know that:
tan θ = |(m2 - m1) / (1 + m1m2)|
tan θ = |(3 - 1 / 2) / (1 + (1 / 2) × 3)|
tan θ = |(5 / 2) / (1 + 3 / 2)|
tan θ = |(5 / 2) / (5 / 2)|
tan θ = 1
Therefore,
θ = 45°
Answer:
45°
Example 17
Question in English:
If A(-2, 1), B(2, 3), and C(-2, -4) are the vertices of triangle ABC, show that:
tan B = 2 / 3
Question in Hindi:
यदि A(-2, 1), B(2, 3), और C(-2, -4) त्रिभुज ABC के शीर्ष हैं, तो सिद्ध कीजिए:
tan B = 2 / 3
Solution:
Clearly, angle B is the angle between BA and BC.
Slope of BA:
m1 = (3 - 1) / (2 - (-2))
m1 = 2 / 4
m1 = 1 / 2
Slope of BC:
m2 = (-4 - 3) / (-2 - 2)
m2 = -7 / -4
m2 = 7 / 4
Now,
tan B = |(m2 - m1) / (1 + m1m2)|
tan B = |((7 / 4) - (1 / 2)) / (1 + (7 / 4)(1 / 2))|
tan B = |((7 / 4) - (2 / 4)) / (1 + 7 / 8)|
tan B = |(5 / 4) / (15 / 8)|
tan B = (5 / 4) × (8 / 15)
tan B = 2 / 3
Hence proved.
Example 18
Question in English:
If the angle between two lines is π / 4 and the slope of one of the lines is 1 / 2, find the slope of the other line.
Question in Hindi:
यदि दो रेखाओं के बीच का कोण π / 4 है और उनमें से एक रेखा की ढाल 1 / 2 है, तो दूसरी रेखा की ढाल ज्ञात कीजिए।
Solution:
Let:
m1 = 1 / 2
m2 = m
θ = π / 4
We know that:
tan θ = |(m2 - m1) / (1 + m1m2)|
Therefore,
tan (π / 4) = |(m - 1 / 2) / (1 + m / 2)|
1 = |(m - 1 / 2) / (1 + m / 2)|
1 = |(2m - 1) / (2 + m)|
So,
(2m - 1) / (2 + m) = 1
or
(2m - 1) / (2 + m) = -1
Case 1:
(2m - 1) / (2 + m) = 1
2m - 1 = 2 + m
m = 3
Case 2:
(2m - 1) / (2 + m) = -1
2m - 1 = -(2 + m)
2m - 1 = -2 - m
3m = -1
m = -1 / 3
Answer:
m = 3 or m = -1 / 3
Exercise
Question 1
English:
Find the slope of a line whose inclination is:
(i) 30°
(ii) 120°
(iii) 135°
(iv) 90°
Hindi:
उस रेखा की ढाल ज्ञात कीजिए जिसका झुकाव है:
(i) 30°
(ii) 120°
(iii) 135°
(iv) 90°
Answers:
(i) 1 / √3
(ii) -√3
(iii) -1
(iv) Not defined
Question 2
English:
Find the inclination of a line whose slope is:
(i) √3
(ii) 1 / √3
(iii) 1
(iv) -1
(v) -√3
Hindi:
उस रेखा का झुकाव ज्ञात कीजिए जिसकी ढाल है:
(i) √3
(ii) 1 / √3
(iii) 1
(iv) -1
(v) -√3
Answers:
(i) 60°
(ii) 30°
(iii) 45°
(iv) 135°
(v) 120°
Question 3
English:
Find the slope of a line which passes through the points:
(i) (0, 0) and (4, -2)
(ii) (0, -3) and (2, 1)
(iii) (2, 5) and (-4, -4)
(iv) (-2, 3) and (4, -6)
Hindi:
उस रेखा की ढाल ज्ञात कीजिए जो निम्नलिखित बिंदुओं से होकर गुजरती है:
(i) (0, 0) और (4, -2)
(ii) (0, -3) और (2, 1)
(iii) (2, 5) और (-4, -4)
(iv) (-2, 3) और (4, -6)
Answers:
(i) -1 / 2
(ii) 2
(iii) 3 / 2
(iv) -3 / 2
Question 4
English:
If the slope of the line joining the points A(x, 2) and B(6, -8) is -5 / 4, find the value of x.
Hindi:
यदि बिंदु A(x, 2) और B(6, -8) को मिलाने वाली रेखा की ढाल -5 / 4 है, तो x का मान ज्ञात कीजिए।
Answer:
x = -2
Question 5
English:
Show that the line through the points (5, 6) and (2, 3) is parallel to the line through the points (9, -2) and (6, -5).
Hindi:
सिद्ध कीजिए कि (5, 6) और (2, 3) से गुजरने वाली रेखा, (9, -2) और (6, -5) से गुजरने वाली रेखा के समांतर है।
Question 6
English:
Find the value of x so that the line through (3, x) and (2, 7) is parallel to the line through (-1, 4) and (0, 6).
Hindi:
x का मान ज्ञात कीजिए ताकि (3, x) और (2, 7) से गुजरने वाली रेखा, (-1, 4) और (0, 6) से गुजरने वाली रेखा के समांतर हो।
Answer:
x = 9
Question 7
English:
Show that the line through the points (-2, 6) and (4, 8) is perpendicular to the line through the points (3, -3) and (5, -9).
Hindi:
सिद्ध कीजिए कि (-2, 6) और (4, 8) से गुजरने वाली रेखा, (3, -3) और (5, -9) से गुजरने वाली रेखा पर लंब है।
Question 8
English:
If A(2, -5), B(-2, 5), C(x, 3) and D(1, 1) are four points such that AB and CD are perpendicular to each other, find the value of x.
Hindi:
यदि A(2, -5), B(-2, 5), C(x, 3) और D(1, 1) ऐसे चार बिंदु हैं कि AB और CD एक-दूसरे पर लंब हैं, तो x का मान ज्ञात कीजिए।
Answer:
x = 6
Question 9
English:
Without using Pythagoras theorem, show that the points A(1, 2), B(4, 5) and C(6, 3) are the vertices of a right-angled triangle.
Hindi:
पाइथागोरस प्रमेय का प्रयोग किए बिना सिद्ध कीजिए कि बिंदु A(1, 2), B(4, 5) और C(6, 3) एक समकोण त्रिभुज के शीर्ष हैं।
Question 10
English:
Using slopes, show that the points A(6, -1), B(5, 0) and C(2, 3) are collinear.
Hindi:
ढालों का प्रयोग करके सिद्ध कीजिए कि बिंदु A(6, -1), B(5, 0) और C(2, 3) संरेखीय हैं।
Question 11
English:
Using slopes, find the value of x for which the points A(5, 1), B(1, -1) and C(x, 4) are collinear.
Hindi:
ढालों का प्रयोग करके x का मान ज्ञात कीजिए जिसके लिए बिंदु A(5, 1), B(1, -1) और C(x, 4) संरेखीय हों।
Answer:
x = 11
Question 12
English:
Using slopes, show that the points A(-4, -1), B(-2, -4), C(4, 0) and D(2, 3) taken in order, are the vertices of a rectangle.
Hindi:
ढालों का प्रयोग करके सिद्ध कीजिए कि बिंदु A(-4, -1), B(-2, -4), C(4, 0) और D(2, 3) क्रम में लेने पर एक आयत के शीर्ष हैं।
Question 13
English:
Using slopes, prove that the points A(-2, -1), B(1, 0), C(4, 3) and D(1, 2) are the vertices of a parallelogram.
Hindi:
ढालों का प्रयोग करके सिद्ध कीजिए कि बिंदु A(-2, -1), B(1, 0), C(4, 3) और D(1, 2) एक समांतर चतुर्भुज के शीर्ष हैं।
Question 14
English:
If the three points A(h, k), B(x1, y1) and C(x2, y2) lie on a line, then show that:
(h - x1)(y2 - y1) = (k - y1)(x2 - x1)
Hindi:
यदि तीन बिंदु A(h, k), B(x1, y1) और C(x2, y2) एक ही रेखा पर स्थित हैं, तो सिद्ध कीजिए:
(h - x1)(y2 - y1) = (k - y1)(x2 - x1)
Question 15
English:
If the points A(a, 0), B(0, b) and P(x, y) are collinear, using slopes, prove that:
x / a + y / b = 1
Hindi:
यदि बिंदु A(a, 0), B(0, b) और P(x, y) संरेखीय हैं, तो ढालों का प्रयोग करके सिद्ध कीजिए:
x / a + y / b = 1
Question 16
English:
A line passes through the points A(4, -6) and B(-2, -5). Show that the line AB makes an obtuse angle with the x-axis.
Hindi:
एक रेखा बिंदु A(4, -6) और B(-2, -5) से होकर गुजरती है। सिद्ध कीजिए कि रेखा AB x-अक्ष के साथ अधिककोण बनाती है।
Question 17
English:
The vertices of a quadrilateral are A(-4, 2), B(2, 6), C(8, 5) and D(9, -7). Using slopes, show that the midpoints of the sides of quadrilateral ABCD form a parallelogram.
Hindi:
चतुर्भुज के शीर्ष A(-4, 2), B(2, 6), C(8, 5) और D(9, -7) हैं। ढालों का प्रयोग करके सिद्ध कीजिए कि चतुर्भुज ABCD की भुजाओं के मध्यबिंदु एक समांतर चतुर्भुज बनाते हैं।
Question 18
English:
Find the slope of the line which makes an angle of 30° with the positive direction of the y-axis, measured anticlockwise.
Hindi:
उस रेखा की ढाल ज्ञात कीजिए जो y-अक्ष की धनात्मक दिशा के साथ वामावर्त दिशा में 30° का कोण बनाती है।
Answer:
-√3
Question 19
English:
Find the angle between the lines whose slopes are √3 and 1 / √3.
Hindi:
उन रेखाओं के बीच का कोण ज्ञात कीजिए जिनकी ढालें √3 और 1 / √3 हैं।
Answer:
30°
Question 20
English:
Find the angle between the lines whose slopes are (2 - √3) and (2 + √3).
Hindi:
उन रेखाओं के बीच का कोण ज्ञात कीजिए जिनकी ढालें (2 - √3) और (2 + √3) हैं।
Answer:
60°
Question 21
English:
If A(1, 2), B(-3, 2) and C(3, -2) are the vertices of triangle ABC, show that:
(i) tan A = 2
(ii) tan B = 2 / 3
(iii) tan C = 4 / 7
Hindi:
यदि A(1, 2), B(-3, 2) और C(3, -2) त्रिभुज ABC के शीर्ष हैं, तो सिद्ध कीजिए:
(i) tan A = 2
(ii) tan B = 2 / 3
(iii) tan C = 4 / 7
Question 22
English:
If θ is the angle between the lines joining the points A(0, 0) and B(2, 3), and the points C(2, -2) and D(3, 5), show that:
tan θ = 11 / 23
Hindi:
यदि θ उन रेखाओं के बीच का कोण है जो बिंदु A(0, 0), B(2, 3) तथा C(2, -2), D(3, 5) को मिलाती हैं, तो सिद्ध कीजिए:
tan θ = 11 / 23
Question 23
English:
If θ is the angle between the diagonals of a parallelogram ABCD whose vertices are A(0, 2), B(2, -1), C(4, 0) and D(2, 3), show that:
tan θ = 2
Hindi:
यदि θ समांतर चतुर्भुज ABCD के विकर्णों के बीच का कोण है जिसके शीर्ष A(0, 2), B(2, -1), C(4, 0) और D(2, 3) हैं, तो सिद्ध कीजिए:
tan θ = 2
Question 24
English:
Show that the points A(0, 6), B(2, 1) and C(7, 3) are three corners of a square ABCD. Find:
(i) the slope of the diagonal BD
(ii) the coordinates of the fourth vertex D
Hindi:
सिद्ध कीजिए कि बिंदु A(0, 6), B(2, 1) और C(7, 3) वर्ग ABCD के तीन शीर्ष हैं। ज्ञात कीजिए:
(i) विकर्ण BD की ढाल
(ii) चौथे शीर्ष D के निर्देशांक
Answers:
(i) 7 / 3
(ii) (5, 8)
Question 25
English:
A(1, 1), B(7, 3) and C(3, 6) are the vertices of triangle ABC. If D is the midpoint of BC and AL ⟂ BC, find the slopes of:
(i) AD
(ii) AL
Hindi:
A(1, 1), B(7, 3) और C(3, 6) त्रिभुज ABC के शीर्ष हैं। यदि D, BC का मध्यबिंदु है तथा AL ⟂ BC है, तो निम्नलिखित की ढाल ज्ञात कीजिए:
(i) AD
(ii) AL
Answers:
(i) 7 / 8
(ii) 4 / 3