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# If ∆ PQR and ∆ PRS are equilateral, what fraction of PQRS is shaded?

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Joined: 02 Sep 2009
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If ∆ PQR and ∆ PRS are equilateral, what fraction of PQRS is shaded?  [#permalink]

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22 Sep 2017, 00:32
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75% (01:34) correct 25% (02:01) wrong based on 73 sessions

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If ∆ PQR and ∆ PRS are equilateral, what fraction of PQRS is shaded?

(A) 1/3
(B) 1/4
(C) 1/6
(D) 1/9
(E) 1/12

Attachment:

2017-09-20_1023.png [ 10.15 KiB | Viewed 1477 times ]

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Posts: 6808
Re: If ∆ PQR and ∆ PRS are equilateral, what fraction of PQRS is shaded?  [#permalink]

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22 Sep 2017, 03:19
Bunuel wrote:

If ∆ PQR and ∆ PRS are equilateral, what fraction of PQRS is shaded?

(A) 1/3
(B) 1/4
(C) 1/6
(D) 1/9
(E) 1/12

Attachment:
2017-09-20_1023.png

Hi...
∆PQR and ∆ PRS are equilateral and same as they share common side PR..
Concentrate now on ∆PRS..
The altitude from the vertices intersect at 1/3 of altitude/height.
So the unshaded triangles along PR gave 1/3 the area of ∆PRS.
The remaining portion is 2/3 of area.
AND it can be seen that the shaded portion is 1/2 of this so 1/2 * 2/3 =1/3 of ∆PRS..
But quad PQRS = PRS + PQS=2 times area of PRS

So shaded portion is 1/3 * 1/2= 1/6 of PQRS

C
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If ∆ PQR and ∆ PRS are equilateral, what fraction of PQRS is shaded?  [#permalink]

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22 Sep 2017, 10:14
1
Bunuel wrote:
If ∆ PQR and ∆ PRS are equilateral, what fraction of PQRS is shaded?

(A) 1/3
(B) 1/4
(C) 1/6
(D) 1/9
(E) 1/12

Attachment:
The attachment 2017-09-20_1023.png is no longer available

Attachment:

equitriangleshaded.png [ 13.32 KiB | Viewed 1114 times ]

Finding the answer to this question did not take a ton of time, but explaining that answer without a diagram is hard

1) SIX CONGRUENT SMALL TRIANGLES

Consider just ∆ PRS in figure PQRS

An equilateral triangle's altitude is also its median, angle bisector, and perpendicular bisector of base

All medians of an equilateral triangle are the same length:
Medians bisect the 60-degree angles to form two 30-degree angles
Medians bisect the bases into equal lengths and create right angles

Medians, therefore, create six identical 30-60-90 triangles

2) ANGLE MEASURES AND SIDE LENGTHS OF SIX TRIANGLES ARE CONGRUENT

Each triangle has:
60-degree angle at circumcenter, 90-degree angle at base bisector point, 30-degree angle at vertex

30-60-90 triangles have sides in ratio: $$x: x\sqrt{3}: 2x$$

See angles and side lengths in diagram of each small triangle. They are identical.

3) SHADED PORTION = $$\frac{1}{3}$$OF ONE TRIANGLE and $$\frac{1}{6}$$ of figure PQRS

Because the six small triangles are equal,
one shaded portion equals $$\frac{1}{6}$$ of area of ∆ PRS

There are two shaded portions. $$\frac{1}{6} + \frac{1}{6} = \frac{1}{3}$$ of area of ∆ PRS

Given: ∆ PQR and ∆ PRS are equilateral
Length of PR = length of both PQ and QR
Areas, therefore, of ∆ PQR and ∆ PRS are equal

∆ PRS is $$\frac{1}{2}$$ of the area of figure PQRS

Find shaded area's fraction of ∆ PRS and double the fraction

$$\frac{1}{3} * \frac{1}{2} = \frac{1}{6}$$of the area of PQRS

Fraction of PQRS that is shaded = $$\frac{1}{6}$$

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Re: If ∆ PQR and ∆ PRS are equilateral, what fraction of PQRS is shaded?  [#permalink]

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22 Sep 2017, 13:40
This is a good question.
All one has to do is assume the area of the small triangle as "a" and after that, we just need to jot all the areas and find the fraction...

2a/12a = 1/6
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If ∆ PQR and ∆ PRS are equilateral, what fraction of PQRS is shaded?  [#permalink]

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22 Sep 2017, 14:17
Bunuel wrote:

If ∆ PQR and ∆ PRS are equilateral, what fraction of PQRS is shaded?

(A) 1/3
(B) 1/4
(C) 1/6
(D) 1/9
(E) 1/12

Attachment:
2017-09-20_1023.png

Since both triangle PQR and PRS are equilateral, we can easily assume that the perpendicular drawn from one vertex will also be the angle bisector for that vertex as well as perpendicular bisector for the opposite side. This way 3 perpendicular bisectors create 6 congruent triangles.
Let area of triangle PRS be x, then area of each shaded triangle will be x/6. Hence area of 2 shaded region will be 2*x/6 = x/3.
Since Triangle PQR and PRS are equilateral triangle sharing one side, hence these 2 are also congruent. This means area of triangle PQR = x
hence fraction is (x/3)/(x+x) ==>(x/3)/2x ==> 1/6

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Re: If ∆ PQR and ∆ PRS are equilateral, what fraction of PQRS is shaded?  [#permalink]

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10 Jun 2018, 05:18
Bunuel wrote:
If ∆ PQR and ∆ PRS are equilateral, what fraction of PQRS is shaded?

(A) 1/3
(B) 1/4
(C) 1/6
(D) 1/9
(E) 1/12

Attachment:
2017-09-20_1023.png

Consider triangle PRS:
Area of triangle PRS = 1/2 * PS * RD (suppose is the point of intersection of the perpendicular drawn from R to PS )

Now, all the altitudes intersect at the same point O and divide one another in the ratio 2:1.
Therefore, OD = 1/3 * RD
PD = 1/2 * PS
Therefore, area of shaded region ODP = 1/2 * 1/3 RD * 1/2 PS = area of shaded region ROE

Total area of PQRS = 2 * area of PRS (because triangle PRS and PQR are same as they share same side PR and are equilateral)
= 2 * 1/2 * PS * RD
Total area of shaded regions = 2 * 1/2 * 1/3 RD * 1/2 PS

Therefore, Area of shaded region : area of PQRS = 1/6
Re: If ∆ PQR and ∆ PRS are equilateral, what fraction of PQRS is shaded? &nbs [#permalink] 10 Jun 2018, 05:18
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# If ∆ PQR and ∆ PRS are equilateral, what fraction of PQRS is shaded?

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