Question

a,b,c,d are in proportion.If b+c=7,a+d=8 and a² +b²+c²+d²=65,then find the values of a,b,c and d​

Answer 1

Answer:

If b : a = c : d, then a, b, c, d are in proportion. Four quantities are said to be in proportion, if the ratio of the first and the second quantities is equal to the ratio of the third and the fourth quantities.

Step-by-step explanation:

Step: 1  A, B, C, and D are in proportion if b: a = c: d. If the ratio between the first and second amounts is the same as the ratio between the third and fourth quantities, then four quantities are said to be in proportion.

Step : 2  a = 55° (Vertically Opposite Angles are Equal). a + d = 180° (Co - interior Angles are Supplementary). c + d = 180° (Co - interior Angles are Supplementary).

Step : 3  a,b,c,d are in proportion

$\begin{aligned}& \frac{a}{b}=\frac{c}{d}=\mathrm{k} \text { (say) } \\& \mathrm{a}=\mathrm{bk}, \mathrm{c}=\mathrm{dk} \\& \text { L.H.S. }=a b c d\left(\frac{1}{a^2}+\frac{1}{b^2}+\frac{1}{c^2}+\frac{1}{d^2}\right)\end{aligned}$

$\begin{aligned}& =b k \cdot b \cdot d k \cdot d\left[\frac{1}{b^2 k^2}+\frac{1}{b^2}+\frac{1}{d^2 k^2}+\frac{1}{d^2}\right] \\& =k^2 b^2 d^2\left[\frac{d^2+d^2 k^2+b^2+b^2 k^2}{b^2 d^2 k^2}\right]\end{aligned}$

$\begin{aligned}& =d^2\left(1+k^2\right)+b^2(1+k 2) \\& =\left(1+k^2\right)\left(b^2+d^2\right) \\& \text { R.H.S. }=a^2+b^2+c 2+d^2 \\& =b^2 k^2+b^2+d^2 k^2+d^2 \\& =b^2\left(k^2+1\right)+d^2\left(k^2+1\right) \\& =\left(k^2+1\right)\left(b^2+d^2\right) \\& \therefore \text { L.H.S. = R.H.S. }\end{aligned}$

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Answer 2

Step-by-step explanation:

follow above method need to focus on a2 + b2 +c2 +d2 =65