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Two equal right-angled triangles of height 12 cm and base 5 cm are cut out from an equilateral triangle as shown in the figure below, such that the area of the unshaded region and the area of the shaded region are in the ratio (5 – √3) ∶ √3. Find the perimeter of the remaining figure.

Two equal right-angled triangles of height 12 cm and base 5 cm are cut out from an equilateral triangle as shown in the figure below, such that the area of the unshaded region and the area of the shaded region are in the ratio (5 – √3) ∶ √3. Find the perimeter of the remaining figure. Correct Answer 68 cm

Given:

Height and base of right angle triangle = 12 cm and 5 cm

Formula used:

Area of triangle = 1/2 × base × height

Area of equilateral triangle = √3/4 × (side)2

Calculation:

Given,

Base of right-angled triangle = 5 cm

Height of right-angled triangle = 12 cm

⇒ Using Pythagoras theorem, Hypotenuse of right-angled triangle = √(52 + 122) = √169 = 13 cm

∵ Area of right-angled triangle = 1/2 × base × height

⇒ Area of shaded region = 2 × 1/2 × 5 × 12 = 60 cm2

Also,

Area of unshaded region/Area of shaded region = (5 – √3)/√3

⇒ Area of unshaded region = (5/√3 – 1) × Area of shaded region

⇒ Area of unshaded region + Area of shaded region = (5/√3) × Area of shaded region

⇒ Area of equilateral triangle = (5/√3) × 60 = 100√3 cm2

∵ Area of equilateral triangle = (√3/4) × (side)2

⇒ Side of equilateral triangle = √400 = 20 cm

Now,

Perimeter of remaining figure = Perimeter of equilateral triangle + 2 (height of right triangle + base of right triangle – hypotenuse of right triangle)

⇒ Perimeter of remaining figure = 3 × 20 + 2(12 + 5 – 13) = 60 + 8 = 68 cm

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