Question
If $(a + b + c)(a – b + c) = a2 + b2 + c2$, show that $a, b, c$ are in continued proportion.
✓
Answer
$(a + b + c)(a – b + c) = a^2 + b^2 + c^2 …$[Given]
$\therefore a(a – b + c) + b(a – b + c) + c(a – b + c) = a2 + b2 + c2$
$\therefore a^2 – ab + ac + ab – b^2 + be + ac – be + c^2 = a^2 + b^2 + c^2$
$\therefore a^2 + 2ac – b^2 + c^2 = a^2 + b^2 + c^2$
$\therefore 2ac – b^2 = b^2$
$\therefore 2ac = 2b^2$
$\therefore ac = b^2$
$\therefore b^2 = ac$
$\therefore a, b, c$ are in continued proportion.
$\therefore a(a – b + c) + b(a – b + c) + c(a – b + c) = a2 + b2 + c2$
$\therefore a^2 – ab + ac + ab – b^2 + be + ac – be + c^2 = a^2 + b^2 + c^2$
$\therefore a^2 + 2ac – b^2 + c^2 = a^2 + b^2 + c^2$
$\therefore 2ac – b^2 = b^2$
$\therefore 2ac = 2b^2$
$\therefore ac = b^2$
$\therefore b^2 = ac$
$\therefore a, b, c$ are in continued proportion.
Need a full question paper?
Generate a complete, print-ready paper with questions like this in minutes — across 16+ boards, with answer keys.
Explore more
Similar questions
Using the property $\frac{a}{b}=\frac{a k}{b k}$, fill in the blanks by substituting proper numbers in the following.
i. $\frac{5}{7}=\frac{\ldots .}{28}=\frac{35}{\ldots . .}=\frac{\ldots}{3.5}$
ii. $\quad \frac{9}{14}=\frac{4.5}{\ldots . .}=\frac{\ldots . .}{42}=\frac{\ldots .}{3.5}$
→i. $\frac{5}{7}=\frac{\ldots .}{28}=\frac{35}{\ldots . .}=\frac{\ldots}{3.5}$
ii. $\quad \frac{9}{14}=\frac{4.5}{\ldots . .}=\frac{\ldots . .}{42}=\frac{\ldots .}{3.5}$
Construct ∆PQR such that ∠P = 70°, ∠R = 50°, QR = 7.3 cm and construct its circumcircle.
In a competitive examination, there were 60 questions. The correct answer would carry 2 marks, and for incorrect answer 1 mark would be subtracted. Yashwant had attempted all the questions and he got total 90 marks. Then how many questions he got wrong?
Take the set of natural numbers from 1 to 20 as universal set and show set X and Y using Venn diagram. [2 Marks each]
i. X= {x |x ∈ N, and 7 < x < 15}
ii. Y = { y | y ∈ N, y is a prime number from 1 to 20}
i. X= {x |x ∈ N, and 7 < x < 15}
ii. Y = { y | y ∈ N, y is a prime number from 1 to 20}
In a $\triangle\text{ABC},$ the internal bisectors of $\angle\text{B}$ and $\angle\text{E}$ meet at P and the external bisectors of $\angle\text{B}$ and $\angle\text{C}$ meet at Q. Prove that $\angle\text{BPC}+\angle\text{BQC}=180^\circ.$
→On a number line, co-ordinates of P, Q, R are 3,-5 and 6 respectively. State with reason whether the following statements are true or false.
i. d(p, Q) + d(Q, R) = d(P, R)
ii. d(P, R) + d(R, Q) = d(P, Q)
iii. d(R, P) + d(P, Q) = d(R, Q)
iv. d(P, Q) – d(P, R) = d(Q, R)
i. d(p, Q) + d(Q, R) = d(P, R)
ii. d(P, R) + d(R, Q) = d(P, Q)
iii. d(R, P) + d(P, Q) = d(R, Q)
iv. d(P, Q) – d(P, R) = d(Q, R)
Construct ∆ABC such that ∠B =100°, BC = 6.4 cm, ∠C = 50° and construct its incircle.
In the given figure, three lines AB, CD and EF intersect at a point O such that $\angle\text{AOE}=35^\circ$ and $\angle\text{BOD}=40^\circ.$ Find the measure of $\angle\text{AOC},\angle\text{BOF},\angle\text{COF}$ and $\angle\text{DOE}.$
→What must be subtracted from $(x^4 + 2x^3 - 2x^2 + 4x + 6)$ so that the result is exactly divisible by $(x^2 + 2x - 3)$?
→Simplify the following products:$\Big(\frac{\text{x}}{2}-\frac{2}{5}\Big)\Big(\frac{2}{5}-\frac{\text{x}}{2}\Big)-\text{x}^2+2\text{x}$
→