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 r_{1} + r_{2} + r_{3} – r = 4R.

Solution:

Question 4.

In ∆ABC, prove that r + r_{1} + r_{2} – r_{3} = 4R cos C.

Solution:

Question 5.

If r + r_{1} + r_{2} + r_{3} then show that C = 90°.

Solution:

II.

Question 1.

Prove that 4(r_{1}r_{2} + r_{2}r_{3} + r_{3}r_{1}) = (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(r_{1} + r_{2} + r_{3}) = ab + bc + ca – s^{2}.

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 r_{1}r_{2}r_{3} = \(r^{3} \cot ^{2} \frac{A}{2} \cdot \cot ^{2} \frac{B}{2} \cdot \cot ^{2} \frac{C}{2}\)

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 = (r_{2} + r_{3}) \(\sqrt{\frac{r r_{1}}{r_{2} r_{3}}}\)

(ii) ∆ = r_{1}r_{2} \(\sqrt{\frac{4 R-r_{1}-r_{2}}{r_{1}+r_{2}}}\)

Solution:

Question 5.

Prove that \(r_{1}^{2}+r_{2}^{2}+r_{3}^{2}+r^{2}\) = 16R^{2} – (a^{2} + b^{2} + c^{2}).

Solution:

Question 6.

If p_{1}, p_{2}, p_{3} 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) p_{1} . p_{2} . p_{3} = \(\frac{(a b c)^{2}}{8 R^{3}}=\frac{8 \Delta^{3}}{a b c}\)

Solution:

Question 7.

If a = 13, b = 14, c = 15, show that R = \(\frac{65}{8}\), r = 4, r_{1} = \(\frac{21}{2}\), r_{2} = 12 and r_{3} = 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 r_{1} = 2, r_{2} = 3, r_{3} = 6 and r = 1, prove that a = 3, b = 4 and c = 5.

Solution: