Question

Triangle ABC and triangle DEF are equilateral triangles, A(triangle ABC): A(triangle DEF)=3:4. If DE=4 then what is length of AB?​

Answer 1

★Given:-

• ΔABC &  ΔDEF are equilateral triangles.
• Ar.(ΔABC):Ar.Δ(DEF) = 3:4

★To find:-

• Length of AB=?

★Solution:-

We now,

For an equilateral triangle,

⋆All the sides are equal.

$\leadsto \underline{\boxed{\sf Area=\frac{\sqrt{3} }{4} a^2 }}$

Where a = side.

Now,

$:\implies \sf \frac{Ar.(\triangle ABC )}{Ar.(\triangle DE F )} = \frac{3}{4} \\\\:\implies \sf \frac{\frac{\sqrt{3} }{4}(AB)^2 }{\frac{\sqrt{3} }{4} (DE)^2} =\frac{3}{4} \\\\:\implies \sf \frac{(AB)^2}{(DE)^2} =\frac{3}{4} \\\\:\implies \sf \frac{AB}{DE} =\sqrt{\frac{3}{4} }$

Given DE = 4,so,

$:\implies \sf AB = 4(\sqrt{\frac{3}{4}} \ )\\\\:\implies \sf AB = 4(\frac{\sqrt{3} }{2})\\\\:\implies \boxed{\boxed{\sf AB = 2\sqrt{3} }}$

Hence,

The length of AB = 2√3.

-------------------------

Know more:-

✦Equilateral triangle :

• A triangle in which all the sides are equal & measure of all the internal angles is 60° is an equilateral triangle.
• Perimeter = 3a.
• The ortho-center and centroid lies on the same point.

✦If two triangles  are similar,the ratio of the area = square to the ratio of the corresponding sides.

✦Scalene triangle :

• A triangle in which all sides are different.
• All the internal angles are also different.

✦Isosceles triangle:

• A triangle in which two sides are equal and one side is different.
• And two angles are different.

_______________

Answer 2

★Given:-

ΔABC &  ΔDEF are equilateral triangles.

Ar.(ΔABC):Ar.Δ(DEF) = 3:4

★To find:-

Length of AB=?

★Solution:-

We now,

For an equilateral triangle,

⋆All the sides are equal.

$\leadsto \underline{\boxed{\sf Area=\frac{\sqrt{3} }{4} a^2 }}$

Where a = side.

Now,

$:\implies \sf \frac{Ar.(\triangle ABC )}{Ar.(\triangle DE F )} = \frac{3}{4} \\\\:\implies \sf \frac{\frac{\sqrt{3} }{4}(AB)^2 }{\frac{\sqrt{3} }{4} (DE)^2} =\frac{3}{4} \\\\:\implies \sf \frac{(AB)^2}{(DE)^2} =\frac{3}{4} \\\\:\implies \sf \frac{AB}{DE} =\sqrt{\frac{3}{4} }$

Given DE = 4,so,

$:\implies \sf AB = 4(\sqrt{\frac{3}{4}} \ )\\\\:\implies \sf AB = 4(\frac{\sqrt{3} }{2})\\\\:\implies \boxed{\boxed{\sf AB = 2\sqrt{3} }}$

Hence,

The length of AB = 2√3.

-------------------------

Know more:-

✦Equilateral triangle :

A triangle in which all the sides are equal & measure of all the internal angles is 60° is an equilateral triangle.

Perimeter = 3a.

The ortho-center and centroid lies on the same point.

✦If two triangles  are similar,the ratio of the area = square to the ratio of the corresponding sides.

✦Scalene triangle :

A triangle in which all sides are different.

All the internal angles are also different.

✦Isosceles triangle:

A triangle in which two sides are equal and one side is different.

And two angles are different.

_______________