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# If side BE has length 10 and side AC has length 8, what is the area of

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Math Expert
Joined: 02 Sep 2009
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If side BE has length 10 and side AC has length 8, what is the area of  [#permalink]

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05 Nov 2019, 01:58
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65% (hard)

Question Stats:

64% (02:52) correct 36% (02:59) wrong based on 33 sessions

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If side BE has length 10 and side AC has length 8, what is the area of the triangle BOC ?

A. $$2\sqrt{3}$$
B. $$4\sqrt{3}$$
C. $$6\sqrt{3}$$
D. $$8\sqrt{3}$$
E. $$12\sqrt{3}$$

Are You Up For the Challenge: 700 Level Questions

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triangle BOC.JPG [ 8.97 KiB | Viewed 521 times ]

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Re: If side BE has length 10 and side AC has length 8, what is the area of  [#permalink]

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05 Nov 2019, 05:19
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$$ABC$$ and $$BDE$$ are both $$30-60-90$$ right triangles and so their sides should be in the ratio $$1:\sqrt{3}:2$$

Therefore we have $$AC=8, BC=4, AB=4\sqrt{3}$$ and $$BE=10, DE=5, BD=5\sqrt{3}$$

Draw $$OF$$ perpendicular to $$BD$$

We now have 2 pairs of similar triangles

1. $$ABC$$ & $$OFC$$
2. $$BDE$$ & $$BOF$$

From $$ABC$$ & $$OFC$$
$$\frac{OF}{FC}=\frac{4\sqrt{3}}{4}$$ or $$OF=FC\sqrt{3}$$

From $$BDE$$ & $$BOF$$
$$\frac{OF}{BF}=\frac{5}{5\sqrt{3}}$$ or $$OF=\frac{BF}{\sqrt{3}}$$

So, $$FC\sqrt{3}=\frac{BF}{\sqrt{3}}$$ or $$BF=3FC$$

We also know that $$BF+FC=BC=4$$
So, $$3FC+FC=4$$ and $$FC=1$$

Therefore $$OF=\sqrt{3}$$

Area of $$BOC$$$$=\frac{1}{2}*BC*OF=\frac{1}{2}*4*\sqrt{3} = 2\sqrt{3}$$

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Re: If side BE has length 10 and side AC has length 8, what is the area of  [#permalink]

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05 Nov 2019, 05:39
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Triangle ACB is 30-60-90 triangle

AC=8
hence, BC=8/2=4

Triangle BOC is 30-60-90
Hence area of BOC= $$\frac{1}{2}*2*2\sqrt{3}$$= $$2\sqrt{3}$$

Bunuel wrote:

If side BE has length 10 and side AC has length 8, what is the area of the triangle BOC ?

A. $$2\sqrt{3}$$
B. $$4\sqrt{3}$$
C. $$6\sqrt{3}$$
D. $$8\sqrt{3}$$
E. $$12\sqrt{3}$$

Are You Up For the Challenge: 700 Level Questions

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The attachment triangle BOC.JPG is no longer available

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Untitled.png [ 5.82 KiB | Viewed 404 times ]

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Re: If side BE has length 10 and side AC has length 8, what is the area of  [#permalink]

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11 Nov 2019, 14:42
1
Bunuel wrote:

If side BE has length 10 and side AC has length 8, what is the area of the triangle BOC ?

A. $$2\sqrt{3}$$
B. $$4\sqrt{3}$$
C. $$6\sqrt{3}$$
D. $$8\sqrt{3}$$
E. $$12\sqrt{3}$$

Are You Up For the Challenge: 700 Level Questions

Attachment:
triangle BOC.JPG

We see that triangle BOC is also a 30-60-90 right triangle, as are triangles ABC and BDE. That is because angle ABC is 60 degrees and angle EBD is 30 degrees, which means angle BOC must be 90 degrees.

Since AC = 8 and AC is the hypotenuse of triangle ABC, BC = 4, and AB = 4√3. Similarly, since BC = 4 and BC is the hypotenuse of triangle BOC, CO = 2, and BO = 2√3. Recall that the area of a right triangle is half the product of its legs; thus, the area of triangle BOC is:

1/2 x 2 x 2√3 = 2√3

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Re: If side BE has length 10 and side AC has length 8, what is the area of   [#permalink] 11 Nov 2019, 14:42
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