Question

Side of equilateral triangle PQR is 8 cm then find the area of triangle whose side is half of the side of triangle PQR.

Answer 1

Given: ΔPQR is an equilateral triangle with PQ = QR = PR = 8 cm and ΔABC is an equilateral triangle with AB = BC = AC = 4 cm

To find: A(ΔABC)

Construction: Draw seg AD ⊥ BC, B-D-C

In ΔABD,

∠ADB = 90°   ...[Construction]

∠ABD = 60°   ...[Angle of an equilateral triangle]

∠BAD = 30°   ...[Remaining angle of a triangle]

∴ ΔABD is a 30° – 60° – 90° triangle.

∴ AD = `sqrt(3)/2` AB   ...[Side opposite to 60°]

∴ AD = `sqrt(3)/2 xx 4`

∴ AD = `2sqrt(3)`   ...(i)

Area of triangle = `1/2 xx "base" xx "height"  `

∴ Area of ΔABC = `1/2 xx AD xx BC`

= `1/2 xx 2sqrt(3) xx 4`

= `2 xx 2sqrt(3)`

= `4sqrt(3)`   ...[From (i)]

∴ Area of the triangle whose side is half of the side of ΔPQR is `4sqrt(3)` sq.cm