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What is the area of the triangle $A B C$ if $D$ is the mid-point of $E
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07 Aug 2025, 00:41
07 Aug 2025, 00:41
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What is the area of the triangle $A B C$ if $D$ is the mid-point of $E C$ and the area of the circle having center at the point E is $\(4 \pi\)$ ?
(A) 3
(B) 4
(C) 5
(D) 6
(E) 7
Re: What is the area of the triangle $A B C$ if $D$ is the mid-point of $E
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19 Aug 2025, 10:31
19 Aug 2025, 10:31
Expert Reply
Area of the circle $\(\pi r^2=4 \pi \Rightarrow r=2\)$
As E is the centre of the circle, we get $\\(mathrm{AE}=\mathrm{BE}=\)$ radii of the circle $\(=2\)$.
Also as $D$ is the mid - point of $\(E C\)$, we get $\(E D=D C=2\)$, so the length of $\(A C=A E+E D+D C= .2+2+2=6\)$
Finally the area of triangle $\(\mathrm{ABC}=\frac{1}{2} \times\)$ Base $\(\times\)$ Height $\(=\frac{1}{2} \times \mathrm{AC} \times \mathrm{BE}=\frac{1}{2} \times 6 \times 2=6\)$
Hence the answer is (D).
As E is the centre of the circle, we get $\\(mathrm{AE}=\mathrm{BE}=\)$ radii of the circle $\(=2\)$.
Also as $D$ is the mid - point of $\(E C\)$, we get $\(E D=D C=2\)$, so the length of $\(A C=A E+E D+D C= .2+2+2=6\)$
Finally the area of triangle $\(\mathrm{ABC}=\frac{1}{2} \times\)$ Base $\(\times\)$ Height $\(=\frac{1}{2} \times \mathrm{AC} \times \mathrm{BE}=\frac{1}{2} \times 6 \times 2=6\)$
Hence the answer is (D).
gmatclubot
Re: What is the area of the triangle $A B C$ if $D$ is the mid-point of $E [#permalink]
19 Aug 2025, 10:31
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