Question

What is the length of ab in a rectangle with 3 circles puzzle?

Answer 1

Answer : $\bold{7.35 \: cm}$



Explanation -

See the given attachment.


Finding the radius -

Given - diameter = 3 cm, so radius = 3/2 = 1.5 cm

Given - diameter = 4 cm, so radius = 4/2 = 2 cm

Given - diameter = 6 cm, so radius = 6/2 = 3 cm


Finding the length of MB -

Radius of 2 circles = 3 cm and 2 cm

PS = 2 + 3 = 5 cm.

SR = 3 - 2 = 1 cm .


∆ PRS is a right angle triangle -

Finding the length of PR^2 -

${pr}^{2} = {ps}^{2} - {rs}^{2}$

${pr}^{2} = {5}^{2} - {1}^{2}$

${pr}^{2} = 25 - 1$

${pr}^{2} = 24$

$pr = \sqrt{24}$

$pr = \sqrt[2]{6}$

$so \: the \: length \: of \: MB = \sqrt[2]{6}$

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Finding the length of AM -

Radius of 2 circles = 1.5 cm and 2 cm

OP = 1.5 + 2 = 3.5 cm

PQ = MN - PN - MQ = 6 - 2 - 1.5 = 2.5

PQ = 2.5 cm

∆ OQP is a right angle triangle .

Finding the length of OQ^2 -

${oq}^{2} = {op}^{2} - {pq}^{2}$

${oq}^{2} = {3.5}^{2} - {2.5}^{2}$

${oq}^{2} = 12.25 - 6.25$

${oq}^{2} = 6$

$oq = \sqrt{6}$

$so \: the \: length \: of \: AM \: = \sqrt{6}$

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Finding the length of AB -

AB = MB + AM

$\sqrt[2]{6} = 4.9$

$\sqrt{6} = 2.45$

AB = MB + AM

AB = 4.9 + 2.45

AB = 7.35 cm.


Hence :

$\bold{7.35 \: cm}$

Answer 2

Hope it will be helpful to you