Inter 1st Year Maths 1A Properties of Triangles Solutions Ex 10(b)

Practicing the Intermediate 1st Year Maths 1A Textbook Solutions Inter 1st Year Maths 1A Properties of Triangles Solutions Exercise 10(b) will help students to clear their doubts quickly.

Intermediate 1st Year Maths 1A Properties of Triangles Solutions Exercise 10(b)

All problems in this exercise have reference to ΔABC.

I.

Question 1.
Express \(\Sigma r_{1} \cot \frac{A}{2}\) in terms of s.
Solution:
\(\Sigma r_{1} \cot \frac{A}{2}\) = \(\Sigma\left(s \tan \frac{A}{2}\right) \cot \frac{A}{2}\)
= Σs
= s + s + s
= 3s

Question 2.
Show that Σa cot A = 2(R + r).
Solution:
L.H.S = Σa . cot A
= Σ2R sin A \(\frac{\cos A}{\sin A}\)
= 2R Σ cos A
= \(2 R\left(1+4 \sin \frac{A}{2} \sin \frac{B}{2} \sin \frac{C}{2}\right)\) (From transformants)
= \(2\left(R+4 R \sin \frac{A}{2} \cdot \sin \frac{B}{2} \cdot \sin \frac{C}{2}\right)\)
= 2(R + r)
= R.H.S

Inter 1st Year Maths 1A Properties of Triangles Solutions Ex 10(b)

Question 3.
In ∆ABC, prove that r1 + r2 + r3 – r = 4R.
Solution:
Inter 1st Year Maths 1A Properties of Triangles Solutions Ex 10(b) I Q3

Question 4.
In ∆ABC, prove that r + r1 + r2 – r3 = 4R cos C.
Solution:
Inter 1st Year Maths 1A Properties of Triangles Solutions Ex 10(b) I Q4
Inter 1st Year Maths 1A Properties of Triangles Solutions Ex 10(b) I Q4.1

Question 5.
If r + r1 + r2 + r3 then show that C = 90°.
Solution:
Inter 1st Year Maths 1A Properties of Triangles Solutions Ex 10(b) I Q5

II.

Question 1.
Prove that 4(r1r2 + r2r3 + r3r1) = (a + b + c)2
Solution:
Inter 1st Year Maths 1A Properties of Triangles Solutions Ex 10(b) II Q1

Question 2.
Prove that \(\left(\frac{1}{r}-\frac{1}{r_{1}}\right)\left(\frac{1}{r}-\frac{1}{r_{2}}\right)\left(\frac{1}{r}-\frac{1}{r_{3}}\right)=\frac{a b c}{\Delta^{3}}=\frac{4 R}{r^{2} s^{2}}\)
Solution:
Inter 1st Year Maths 1A Properties of Triangles Solutions Ex 10(b) II Q2

Question 3.
Prove that r(r1 + r2 + r3) = ab + bc + ca – s2.
Solution:
Inter 1st Year Maths 1A Properties of Triangles Solutions Ex 10(b) II Q3

Question 4.
Show that \(\sum \frac{r_{1}}{(s-b)(s-c)}=\frac{3}{r}\)
Solution:
Inter 1st Year Maths 1A Properties of Triangles Solutions Ex 10(b) II Q4

Inter 1st Year Maths 1A Properties of Triangles Solutions Ex 10(b)

Question 5.
Show that \(\left(r_{1}+r_{2}\right) \tan \frac{C}{2}=\left(r_{3}-r\right) \cot \frac{C}{2}=c\)
Solution:
Inter 1st Year Maths 1A Properties of Triangles Solutions Ex 10(b) II Q5

Question 6.
Show that r1r2r3 = \(r^{3} \cot ^{2} \frac{A}{2} \cdot \cot ^{2} \frac{B}{2} \cdot \cot ^{2} \frac{C}{2}\)
Solution:
Inter 1st Year Maths 1A Properties of Triangles Solutions Ex 10(b) II Q6

III.

Question 1.
Show that cos A + cos B + cos C = 1 + \(\frac{r}{R}\)
Solution:
L.H.S = cos A + cos B + cos C
= 2 cos(\(\frac{A+B}{2}\)) cos(\(\frac{A-B}{2}\)) + cos C
Inter 1st Year Maths 1A Properties of Triangles Solutions Ex 10(b) III Q1

Question 2.
Show that \(\cos ^{2} \frac{A}{2}+\cos ^{2} \frac{B}{2}+\cos ^{2} \frac{C}{2}=2+\frac{r}{2 R}\)
Solution:
Inter 1st Year Maths 1A Properties of Triangles Solutions Ex 10(b) III Q2
Inter 1st Year Maths 1A Properties of Triangles Solutions Ex 10(b) III Q2.1

Question 3.
Show that \(\sin ^{2} \frac{A}{2}+\sin ^{2} \frac{B}{2}+\sin ^{2} \frac{C}{2}=1-\frac{r}{2 R}\)
Solution:
Inter 1st Year Maths 1A Properties of Triangles Solutions Ex 10(b) III Q3

Question 4.
Show that
(i) a = (r2 + r3) \(\sqrt{\frac{r r_{1}}{r_{2} r_{3}}}\)
(ii) ∆ = r1r2 \(\sqrt{\frac{4 R-r_{1}-r_{2}}{r_{1}+r_{2}}}\)
Solution:
Inter 1st Year Maths 1A Properties of Triangles Solutions Ex 10(b) III Q4
Inter 1st Year Maths 1A Properties of Triangles Solutions Ex 10(b) III Q4.1

Inter 1st Year Maths 1A Properties of Triangles Solutions Ex 10(b)

Question 5.
Prove that \(r_{1}^{2}+r_{2}^{2}+r_{3}^{2}+r^{2}\) = 16R2 – (a2 + b2 + c2).
Solution:
Inter 1st Year Maths 1A Properties of Triangles Solutions Ex 10(b) III Q5

Question 6.
If p1, p2, p3 are altitudes drawn from vertices A, B, C to the opposite sides of a triangle respectively, then show that
(i) \(\frac{1}{p_{1}}+\frac{1}{p_{2}}+\frac{1}{p_{3}}=\frac{1}{r}\)
(ii) \(\frac{1}{p_{1}}+\frac{1}{p_{2}}-\frac{1}{p_{3}}=\frac{1}{r_{3}}\)
(iii) p1 . p2 . p3 = \(\frac{(a b c)^{2}}{8 R^{3}}=\frac{8 \Delta^{3}}{a b c}\)
Solution:
Inter 1st Year Maths 1A Properties of Triangles Solutions Ex 10(b) III Q6

Question 7.
If a = 13, b = 14, c = 15, show that R = \(\frac{65}{8}\), r = 4, r1 = \(\frac{21}{2}\), r2 = 12 and r3 = 14.
Solution:
a = 13, b = 14, c = 15
s = \(\frac{a+b+c}{2}\)
= \(\frac{13+14+15}{2}\)
= 21
s – a = 21 – 13 = 8
s – b = 21 – 14 = 7
s – c = 21 – 15 = 6
Inter 1st Year Maths 1A Properties of Triangles Solutions Ex 10(b) III Q7

Inter 1st Year Maths 1A Properties of Triangles Solutions Ex 10(b)

Question 8.
If r1 = 2, r2 = 3, r3 = 6 and r = 1, prove that a = 3, b = 4 and c = 5.
Solution:
Inter 1st Year Maths 1A Properties of Triangles Solutions Ex 10(b) III Q8