TS Inter 1st Year Maths 1A Properties of Triangles Important Questions

Students must practice these TS Inter 1st Year Maths 1A Important Questions Chapter 10 Properties of Triangles Important Questions to help strengthen their preparations for exams.

TS Inter 1st Year Maths 1A Properties of Triangles Important Questions

I.

Question 1.
In ΔABC, If a=3,b=4 and sin A=3/4, find the angle B.
Solution :
By sine rule,\(\frac{a}{\sin A}=\frac{b}{\sin B}\)
⇒ sinB =\(\frac{b \sin A}{a}=\frac{4 \times 3 / 4}{3}\) = 1
⇒ sinB = 1 = B = 90°

Question 2.
If the lengths of the sides of a triangle are 3, 4, 5, find the circumradius of the triangle.
Solution:
Since 32+42= 52 the triangle is right angled and hypotenuse = 5 = circum diameter.
∴ Circum radius = \(\frac{5}{2}\) cm.

TS Inter 1st Year Maths 1A Properties of Triangles Important Questions

Question 3.
lf a=6,b=5,c=9, then find the angle A.
Solution:
TS Inter 1st Year Maths 1A Properties of Triangles Important Questions 1

Question 4.
In a Δ ABC, show that ∑(b +c)cosA = 2s
Solution:
LH.S. = ∑ (b + c) cos A
= (b + c) cos A+ (c + a) cos B + (a + b) cos C
= (bcosA+ acos B)+(ccosA + acosC) + (b cos C + c cos B)
= c+b+a
= a+b+c=2s=R.H.S.

Question 5.
In a Δ ABC, if
(a) (a+b+c)(b+c- a) = 3bc, findA.
Solution:
Given (a + b + c)(b + c-a) = 3bc
⇒ (2s) 2(s – a) = 3 bc
TS Inter 1st Year Maths 1A Properties of Triangles Important Questions 2

(b) If a = 4, b = 5, c = 7, find cos \(\frac{B}{2}\)
Solution:
2s = a+b+c = 4+5+7=16
⇒ s = 8
∴ s – b=8-5=3 and
\(\cos \frac{B}{2}=\sqrt{\frac{s(s-b)}{a c}}=\sqrt{\frac{8 \times 3}{4 \times 7}}=\sqrt{\frac{6}{7}}\)

TS Inter 1st Year Maths 1A Properties of Triangles Important Questions

Question 6.
If ΔABC, find \(b \cos ^2 \frac{C}{2}+c \cos ^2 \frac{B}{2}\)
Solution:
TS Inter 1st Year Maths 1A Properties of Triangles Important Questions 3

Question 7.
If \(\tan \frac{\mathrm{A}}{2}=\frac{5}{6}\) and \(\tan \frac{\mathrm{C}}{2}=\frac{2}{5}\)
Solution:
TS Inter 1st Year Maths 1A Properties of Triangles Important Questions 4
⇒ 3s – 3b = s
⇒ 2s = 3b ⇒ a+b +c = 3b ⇒ a = 2b
⇒ a, b, c are in A.P.

Question 8.
If \(\cot \frac{A}{2}=\frac{b+c}{a}\),find angle B
Solution:
\(\cot \frac{A}{2}=\frac{b+c}{a}\)
TS Inter 1st Year Maths 1A Properties of Triangles Important Questions 5

Question 9.
If tan \(\left(\frac{\mathrm{C}-\mathrm{A}}{2}\right)=\mathrm{k} \cot \frac{\mathrm{B}}{2}\),find K
Solution:
Using Napiers rule,
TS Inter 1st Year Maths 1A Properties of Triangles Important Questions 6

Question 10
In ΔABC, Show that \(\frac{b^2-c^2}{a^2}=\frac{\sin (B-C)}{\sin (B+C)}\)
Solution:
TS Inter 1st Year Maths 1A Properties of Triangles Important Questions 7

TS Inter 1st Year Maths 1A Properties of Triangles Important Questions

Question 11.
Show that \((b-c)^2 \cos ^2 \frac{A}{2}+(b+c)^2 \sin ^2 \frac{A}{2}=a^2\)
Solution:
L.H.S= \((b-c)^2 \cos ^2 \frac{A}{2}+(b+c)^2 \sin ^2 \frac{A}{2}\)
TS Inter 1st Year Maths 1A Properties of Triangles Important Questions 8

Question 12.
If ΔABC, Prove that \(\frac{1}{r_1}+\frac{1}{r_2}+\frac{1}{r_3}=\frac{1}{r}\)
Solution:
TS Inter 1st Year Maths 1A Properties of Triangles Important Questions 10

Question 13.
Show that r r1 r2 r3 = Δ2
Solution:
L.H.S. r. r1 . r2 . r3
TS Inter 1st Year Maths 1A Properties of Triangles Important Questions 11

TS Inter 1st Year Maths 1A Properties of Triangles Important Questions

Question 14.
In an equilateral triangle, find the value of  \(\frac{\mathbf{r}}{\mathbf{R}}\)
Solution:
TS Inter 1st Year Maths 1A Properties of Triangles Important Questions 12

Question 15.
The perimeter of ΔABC is 12 cm and it’s in radius is 1 cm. Then find the area of the triangle.
Solution:
Given 2s = 12 ⇒ s = 6 cm. ; r = 1 cm
Area of Δ ABC,
Δ = rs = (1) (6) = 6sq.cm

Question 16.
Show that r r1 = (s – b) (s – c).
Solution:
L.H.S. = r. r1
TS Inter 1st Year Maths 1A Properties of Triangles Important Questions 13

Question 17.
In ΔABC, is = 6sq. cm and s = 1.5cm find ‘r’
Solution:
\(\mathrm{r}=\frac{\Delta}{\mathrm{s}}=\frac{6}{1.5}=\frac{6}{3 / 2}=4 \mathrm{~cm}\)

Question 18.
Show that \(r r_1 \cot \frac{A}{2}=\Delta\)
Solution:
TS Inter 1st Year Maths 1A Properties of Triangles Important Questions 14

TS Inter 1st Year Maths 1A Properties of Triangles Important Questions

II.

Question 1.
In a \(\Delta \mathrm{ABC}\),prove that \(\tan \left(\frac{B-C}{2}\right)=\frac{b-c}{b+c} \cot \frac{A}{2}\)
Solution:
TS Inter 1st Year Maths 1A Properties of Triangles Important Questions 15
TS Inter 1st Year Maths 1A Properties of Triangles Important Questions 16

Question 2.
Prove that cot A+cot B+ cot C \(=\frac{a^2+b^2+c^2}{4 \Delta}\)
Solution:
TS Inter 1st Year Maths 1A Properties of Triangles Important Questions 17
TS Inter 1st Year Maths 1A Properties of Triangles Important Questions 23

TS Inter 1st Year Maths 1A Properties of Triangles Important Questions

Question 3.
Show that r + r3 + r1 – r2 = 4R cos B.
Solution:
TS Inter 1st Year Maths 1A Properties of Triangles Important Questions 18

Question 4.
In a ΔABC, if r1= 8, r2=12, r3 = 24, find a, b, c.
Solution:
TS Inter 1st Year Maths 1A Properties of Triangles Important Questions 19

TS Inter 1st Year Maths 1A Properties of Triangles Important Questions

Question 5.
If the sides of a triangle are 13, 14, 15, then find the circum diameter.
Solution:
Let a= 13,b= 14,c= 15
TS Inter 1st Year Maths 1A Properties of Triangles Important Questions 20

Question 6.
Show that
a2cotA+b2cotB+c2cotC = \(\frac{\mathbf{a b c}}{\mathbf{R}}\)
Solution:
TS Inter 1st Year Maths 1A Properties of Triangles Important Questions 21

Question 7.
Prove that a(b cos C – c cos B) = b2 – c2.
Solution:
TS Inter 1st Year Maths 1A Properties of Triangles Important Questions 22

TS Inter 1st Year Maths 1A Properties of Triangles Important Questions

Question 8.
Show that \(\frac{c-b \cos A}{b-c \cos A}=\frac{\cos B}{\cos C}\)
Solution:
We have c = a cos B + b cos A and
b = a cos C + c cos A
\(\text { L.H.S. }=\frac{c-b \cos A}{b-c \cos A}\)
TS Inter 1st Year Maths 1A Properties of Triangles Important Questions 24

Question 9.
Show that b2 sin 2C + c2sin 2B = 2bc sin A.
Solution:
TS Inter 1st Year Maths 1A Properties of Triangles Important Questions 26

Question 10.
Show that
\(a \cos ^2 \frac{A}{2}+b \cos ^2 \frac{B}{2}+c \cos ^2 \frac{C}{2}=s+\frac{\Delta}{R}\)
Solution:
TS Inter 1st Year Maths 1A Properties of Triangles Important Questions 27
TS Inter 1st Year Maths 1A Properties of Triangles Important Questions 28

Question 11.
In a ΔABC, if acosA=bcosB, prove that the triangle is either isosceles or right angled.
Solution:
a cos A = b cos B
⇒ 2R sin A cos A = 2R sin B cos B
⇒ sin 2A = sin 2B
⇒ sin 2A = sin (180° – 2B)
⇒ 2A = 2B or 2A = 180° – 2B
⇒ A=B or A=90°- B
⇒ A=B or A+B=90°
⇒  c = 90°
∴ The triangle is isosceles or right angled.

TS Inter 1st Year Maths 1A Properties of Triangles Important Questions

Question 12.
If \(\cot \frac{A}{2}: \cot \frac{B}{2}: \cot \frac{C}{2}\) = 3 : 5 : 7, Show that a : b : c =6 : 5 : 4.
Solution:
TS Inter 1st Year Maths 1A Properties of Triangles Important Questions 29
Thens – a = 3ks  – b = 5ks – c= 7k
∴ 3s – (a + b + c) = 15k
= 3s – 2s= 15k=s= 15k
∴ a = s – 3k = a = 12k
b = s – 5k=b= 10k
c = s – 7k c = 8k
∴ a : b : c = 12k : 10k : 8k = 6 : 5 : 4.

Question 13.
Prove that a3cos(B – C)+b3cos(C – A) + cos3 cos (A – B) = 3 abc.
Solution:
∑a3cos(B-C)
= ∑ a2. a cos (B – C)
=∑ a2 2R sin A cos(B – C)
= ∑ a2 2sin(B + C) cos (B – C)
= \(R \Sigma a^2\left(2 \frac{b}{2 R} \cos B+2 \frac{c}{2 R} \cos C\right)\)
= ∑ a2(bcos B +ccosC)
= a2 (bcos B + ccos C) + b2(ccos C + acosA) + c2 (a cos A + b cos B)
= ab (a cos B+ b cos A) + bc (b cos C + c cos B) + ca(ccos A+acos C)
= abc + bca + cab
= 3 abc = R.H.S.

Question 14.
If P1 P2 p3 are the altitudes of a ΔABC to the opposite sides show that
TS Inter 1st Year Maths 1A Properties of Triangles Important Questions 30
Solution:
TS Inter 1st Year Maths 1A Properties of Triangles Important Questions 31
TS Inter 1st Year Maths 1A Properties of Triangles Important Questions 32

TS Inter 1st Year Maths 1A Properties of Triangles Important Questions

Question 15.
The angle of elevation of the top point P of the vertical tower PQ of height h from a point A is 450 and from a point B Is 600, where B is a point at a distance 30 meters from the point A measured along the line AB which makes an angle 30° with AQ. Find the height of the tower.
Solution:
TS Inter 1st Year Maths 1A Properties of Triangles Important Questions 33

Question 16.
Two trees A and B are on the same side of a tiver. From a point C in the river, the distances of the trees A and B are 250 m and 300 m respectively. If the angle C is 450, find the distance between the trees (use \(\sqrt{2}\) = 1.414).
Solution:
TS Inter 1st Year Maths 1A Properties of Triangles Important Questions 34
Given AC = 300 m and BC = 250 m and in the Δ ABC, using cosine rule
AB2 = AC2 + BC2 – 2AC.BC. cos 45°
= (300)2 + (250)2 -2 (300) (250) – \(\frac{1}{\sqrt{2}}\)
= 46450
∴ AB = 215.5 m (approximately)

Question 17.
Express \(\frac{a \cos A+b \cos B+c \cos C}{a+b+c}\) in terms of R and r.
Solution:
TS Inter 1st Year Maths 1A Properties of Triangles Important Questions 35
TS Inter 1st Year Maths 1A Properties of Triangles Important Questions 36

TS Inter 1st Year Maths 1A Properties of Triangles Important Questions

Question 18.
If a = 13, b = 14, c = 15, find r1.
Solution:
\(\frac{\mathrm{a}+\mathrm{b}+\mathrm{c}}{2}=\mathrm{s} \Rightarrow \mathrm{s}=\frac{13+14+15}{2}=\frac{42}{2}=21\)
s – a = 21- 13=8
s – b = 21 – 14 = 7 and
s – c = 21 – 15 = 6
Δ2 = 21 x 8 x 7 6
Δ = 7 x 12 = 84sq. units
r1 = \(\frac{\Delta}{s-a}=\frac{84}{8}\) = 10.5 units

Question 19.
If r r2 = r1 r3, then find B.
Solution:
Given r r2 = r1 r3
TS Inter 1st Year Maths 1A Properties of Triangles Important Questions 37

Question 20.
In a Δ ABC, show that the sides a, b, and c are in A.P. If and only If r1, r2, r3 are in H.P.
Solution :
r1, r2, r3 are in H.P.
TS Inter 1st Year Maths 1A Properties of Triangles Important Questions 38
⇔ s – a, s – b, s – c are in A.P
⇔ – a, – b, – c are in AP.
⇔ a, b, c are in A.P.

Question 21.
If A = 90°, show that 2 (r + R) = b + c
Solution:
L.H.S. = 2 (r + R)
= 2r + 2R
= 2(s – a) tan\(\frac{\mathrm{A}}{2}\) + 2R . 1
= 2 (s – a) tan 45° + 2R sinA (∵ A = 90°)
= (2s – 2a) + a
= 2s – a = a + b + c – a = b + c = R.H.S.

Question 22.
Show that
TS Inter 1st Year Maths 1A Properties of Triangles Important Questions 39
Solution:
TS Inter 1st Year Maths 1A Properties of Triangles Important Questions 40

Question 23.
Prove that Σ (r1 + r) \(\tan \left(\frac{B-C}{2}\right)=0\)
Solution:
TS Inter 1st Year Maths 1A Properties of Triangles Important Questions 41

TS Inter 1st Year Maths 1A Properties of Triangles Important Questions

Question 24.
Show that \(\left(r_1+r_2\right) \sec ^2 \frac{C}{2}=\left(r_2+r_3\right) \sec ^2 \frac{A}{2} =\left(r_3+r_1\right) \sec ^2 \frac{B}{2}\)
Solution:
TS Inter 1st Year Maths 1A Properties of Triangles Important Questions 42
Hence the triangle is right angled at A.

Question 25.
Show that \(\frac{a b-r_1 r_2}{r_3}=\frac{b c-r_2 r_3}{r_1}=\frac{c a-r_3 r_1}{r_2}\)
Solution:
TS Inter 1st Year Maths 1A Properties of Triangles Important Questions 44
TS Inter 1st Year Maths 1A Properties of Triangles Important Questions 45

TS Inter 1st Year Maths 1A Properties of Triangles Important Questions

III.

Question 1.
If A, A1, A2, A3 are the areas of in circle and exercise of a triangle respectively then prove that
\(\frac{1}{\sqrt{A_1}}+\frac{1}{\sqrt{A_2}}+\frac{1}{\sqrt{A_3}}=\frac{1}{\sqrt{A^2}}\)
Solution:
Let r1, r2, r3 and r be the radii of excircies and incircie respectively whose areas are A1, A2, A3 and A.
TS Inter 1st Year Maths 1A Properties of Triangles Important Questions 46

Question 2.
In a Δ ABC prove that
\(\frac{r_1}{b c}+\frac{r_2}{c a}+\frac{r_3}{a b}=\frac{1}{r}-\frac{1}{2 R}\)
Solution:
TS Inter 1st Year Maths 1A Properties of Triangles Important Questions 47
TS Inter 1st Year Maths 1A Properties of Triangles Important Questions 48
TS Inter 1st Year Maths 1A Properties of Triangles Important Questions 49

Question 3.
If (r2 – r1) (r3 – r3) = 2r2r3, show that A=90°
Solution:
Given (r2 – r1) (r3 – r1) = 2r2r3
TS Inter 1st Year Maths 1A Properties of Triangles Important Questions 50

TS Inter 1st Year Maths 1A Properties of Triangles Important Questions

Question 4.
Prove that \(\frac{r_1\left(r_2+r_3\right)}{\sqrt{r_1 r_2+r_2 r_3+r_3 r_1}}=a\)
Solution:
TS Inter 1st Year Maths 1A Properties of Triangles Important Questions 51
TS Inter 1st Year Maths 1A Properties of Triangles Important Questions 52

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