Question

ABC is an equilateral triangle where pqrs is a square in the equilateral triangle ABC and klm is also a equilateral triangle in square pqrs and defg is another square in triangle in triangle klm and ab=bc=cd=1 cm and sides of square pqrs is equal to sides of triangle klm​

Answer 1

✪ Correct Question :- ABC is an equilateral triangle where PQRS is a square with largest area in the equilateral triangle ABC and KSR is also a equilateral triangle in square PQRS and DEFG is another square in triangle. in triangle KSR , sides of square PQRS is equal to sides of triangle KSR. Find the value of x .? ( Side of Square DEFG . ) it is given that, side of Equaliteral ∆ ABC is 1cm.

✰ Formula used :-

→ Side of Square with largest area inside a Equaliteral ∆ with side a is = a /(1+2√3) or = 0.464 a.

✭ Solution :---

Lets Try to Prove the above told formula first .

❁❁ Refer To Image First .. ❁❁

From image we can see that :-

→ Equaliteral ∆ ABC = with side a cm each .

→ A square DEFG inside = with side x cm.

we draw a Perpendicular from vertices A on Square at Point H, that , divide the side of Square in two Equal parts.

Now, in Rt ∆ AHE , we have ,

→ HE = (x/2) .

→ DE is parallel to BC, So, angle AED = angle ACB = 60°..

using Trignometry now,

→ cos(AHE) = HE/AE

→ cos(60°) = (x/2) / AE

→ 1/2 = x/2AE

→ AE = x cm.

Similarly, in Rt∆EFC,

→ sin(60°) = EF/ EC

(As EF is side of square , so x ).

→ (√3/2) = x/EC

→ EC = (2x/√3) cm.

So, we can say that, side of Equaliteral ∆ABC ,

→ AC = AE + EC

→ a = x + (2x/√3)

→ a = (√3x + 2x ) / √3

→ a = x (√3 + 2) /√3

→ x = a * √3 /(2+√3)

( or, if we put √3 as 1.73 , we get , )

→ x = 0.464a .

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Now, Lets Head to the Question ,

we have given , side of Equaliteral ∆ABC is 1cm.

So, above told concept , we can say that,

→ side of Square PQRS = a = 0.464 * 1 = 0.464 cm.

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Now, it is given that, side of Square PQRS is equal to side of Equaliteral ∆KSR inside it ,

So,

➺ Side of Equaliteral ∆KSR = 0.464cm..

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Now, again , we have a Square inside Equaliteral ∆KSR with side as x cm.

So, with same Concept again now, we can say that :-

☛ Side of Square = x = Side of ∆KSR * 0.464

☛ X = (0.464) * 0.464

☛ X = 0.215cm.

Hence, Value of X will be 0.215cm (Approx).

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★★Extra Brainly Knowledge★★

✯✯ Some Properties of Equaliteral Triangle. ✯✯

• An equilateral triangle is a triangle in which all three sides are equal.

• Equilateral triangles are also equiangular, which means, all three internal angles are also equal to each other and the only value possible is 60° each.

• It is a regular polygon with 3 sides.

• A triangle is equilateral if and only if the circumcenters of any three of the smaller triangles have the same distance from the centroid.

• A triangle is equilateral if and only if any three of the smaller triangles have either the same perimeter or the same inradius.

• The area of an equilateral triangle is basically the amount of space occupied by an equilateral triangle.

• In an equilateral triangle, the median, angle bisector and perpendicular are all the same and can be simply termed as the perpendicular bisector due to congruence conditions.

• The ortho-centre and centroid of the triangle is the same point.

• In an equilateral triangle, median, angle bisector, and altitude for all sides are all the same and are the lines of symmetry of the equilateral triangle.

• The area of an equilateral triangle is (√3a²)/ 4..

• The perimeter of an equilateral triangle is 3a.

• Height of Equaliteral Triangle is (√3a)/2.

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