﻿ The equilateral triangle ABC has an area of 900m^2 . Points D and E are

#### Multiple Choice Question (MCQ)

The equilateral triangle ABC has an area of 900m^2 . Points D and E are the mid points of the side AB and AC respectively. Find the area of the △DEF .

1. 75

2. ×

64

3. ×

85

4. ×

48

Collected in the board: Triangle

Steven Zheng posted 3 years ago

1. BD^2 = BC^2-DC^2

BD = \sqrt{a^2-(\dfrac{1}{2}a )} = \dfrac{\sqrt{3} }{2} a

\dfrac{EF}{DC} = \dfrac{EB}{BD}

\dfrac{EF}{\dfrac{1}{2}a } = \dfrac{\dfrac{1}{2}a}{ \dfrac{\sqrt{3} }{2} a}

EF = \dfrac{\sqrt{3} }{6}a

FG^2 = EF^2-( \dfrac{1}{2}DE )^2 = \dfrac{1}{12}a^2 - \dfrac{1}{16}a^2

FG = \dfrac{\sqrt{3} }{12} a

S_{△DEF} = \dfrac{1}{2}DE\cdotp FG = \dfrac{1}{2}⋅\dfrac{1}{2}a⋅ \dfrac{\sqrt{3} }{12} a = \dfrac{\sqrt{3} }{48}a^2

S_{△ABC} = \dfrac{1}{2} DB⋅AC =\dfrac{1}{2} ⋅ \dfrac{\sqrt{3} }{2} a ⋅ a = \dfrac{\sqrt{3} }{4}a^2

\dfrac{S_{△ABC} }{S_{△DEF}} =12

S_{△DEF} = \dfrac{900}{12} =75

Steven Zheng posted 3 years ago