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 Question 3.
In ∆ABC, prove that r1 + r2 + r3 – r = 4R.
Solution: Question 4.
In ∆ABC, prove that r + r1 + r2 – r3 = 4R cos C.
Solution:  Question 5.
If r + r1 + r2 + r3 then show that C = 90°.
Solution: II.

Question 1.
Prove that 4(r1r2 + r2r3 + r3r1) = (a + b + c)2
Solution: 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: Question 3.
Prove that r(r1 + r2 + r3) = ab + bc + ca – s2.
Solution: Question 4.
Show that $$\sum \frac{r_{1}}{(s-b)(s-c)}=\frac{3}{r}$$
Solution:  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: 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: 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 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:  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: 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:   Question 5.
Prove that $$r_{1}^{2}+r_{2}^{2}+r_{3}^{2}+r^{2}$$ = 16R2 – (a2 + b2 + c2).
Solution: 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: 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  Question 8.
If r1 = 2, r2 = 3, r3 = 6 and r = 1, prove that a = 3, b = 4 and c = 5.
Solution: 