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If a semicircle and triangle are pieced together to form the figure sh

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Bunuel
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If a semicircle and triangle are pieced together to form the figure sh [#permalink] New post   17 Oct 2020, 13:06
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If a semicircle and triangle are pieced together to form the figure shown above, what is the total unit area of the figure?

(1) The length of AB is 2 units.
(2) x = y

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Re: If a semicircle and triangle are pieced together to form the figure sh [#permalink] New post   17 Oct 2020, 22:45
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Bunuel wrote:

If a semicircle and triangle are pieced together to form the figure shown above, what is the total unit area of the figure?

(1) The length of AB is 2 units.
(2) x = y



The area of the semicircle can be found as the radius from the figure is 1. We now need to find the area of the triangle ABC


Statement 1: The length of AB is 2 units.

From this we only get to know that Triangle ABC is isosceles with AB = AC = 2 units.

We cannot find the area of the triangle without knowing the value of the 3rd side (BC)


Therefore Statement 1 Alone is Insufficient. Answer options could be B, C or E



Statement 2: x = y

From this, we get that BC = AC = 2 (Sides opposite equal angles)

We cannot find the area of the triangle without knowing the value of the 3rd side (AB)

Therefore Statement 2 Alone is Insufficient.



Combining Both Statements:

We get that AB = BC = AC = 2. This forms an equilateral triangle and we can find the area of the triangle.


Therefore Both Statements together are sufficient.




Option C

Arun Kumar
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Re: If a semicircle and triangle are pieced together to form the figure sh [#permalink] New post   18 Oct 2020, 06:41
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Bunuel wrote:

If a semicircle and triangle are pieced together to form the figure shown above, what is the total unit area of the figure?

(1) The length of AB is 2 units.
(2) x = y

Show Spoiler
Attachment:
2020-10-18_00-05-22.png


The left side half-circle is fixed, the right side triangle has one side fixed. We need to know 2 more angles on the triangle, 1 angle + 1 length, or 2 more lengths to fix the triangle on the right side.

Statement 1:
We know only 2 lengths now, insufficient.

Statement 2:
x = y which leads to AC = BC = 2 but that's still only two lengths, insufficient.

Combined:
Combined we know all three lengths. The triangle is fixed and we could find the area of it, sufficient.

Ans: C
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If a semicircle and triangle are pieced together to form the figure sh [#permalink] New post   18 Oct 2020, 07:38
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If a semicircle and triangle are pieced together to form the figure shown above, what is the total unit area of the figure?

Step 1: Understanding the question
Area of the semicircle = \(\frac{1}{2} *π r^2 = \frac{1}{2} *π 1^2\) = \(\frac{π}{2}\)

Area of the figure = Area of the semicircle + Area of the triangle =\( \frac{π}{2}\) + Area of the triangle

Step 2: Understanding statement 1 alone
(1) The length of AB is 2 units.
AC = AB = 2, its an isosceles triangle
Area of the triangle cannot be determined
Insufficient

Step 3: Understanding statement 2 alone
(2) x = y
AC = BC, as sides opposite to equal angle are equal.
Area of the triangle cannot be determined
Insufficient

Step 4: Combining statement 1 and 2
AB = BC = CA = 2, its an equilateral triangle
Area of the equilateral triangle can be calculated as the side length is known
Sufficient

C is correct
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Re: If a semicircle and triangle are pieced together to form the figure sh [#permalink] New post   19 Feb 2021, 10:11
CrackVerbalGMAT wrote:
Bunuel wrote:

If a semicircle and triangle are pieced together to form the figure shown above, what is the total unit area of the figure?

(1) The length of AB is 2 units.
(2) x = y



The area of the semicircle can be found as the radius from the figure is 1. We now need to find the area of the triangle ABC


Statement 1: The length of AB is 2 units.

From this we only get to know that Triangle ABC is isosceles with AB = AC = 2 units.

We cannot find the area of the triangle without knowing the value of the 3rd side (BC)


Therefore Statement 1 Alone is Insufficient. Answer options could be B, C or E



Statement 2: x = y

From this, we get that BC = AC = 2 (Sides opposite equal angles)

We cannot find the area of the triangle without knowing the value of the 3rd side (AB)

Therefore Statement 2 Alone is Insufficient.



Combining Both Statements:

We get that AB = BC = AC = 2. This forms an equilateral triangle and we can find the area of the triangle.


Therefore Both Statements together are sufficient.




Option C

Arun Kumar


Hi Arun, I am a bit confused.

So for statement 1, you deduced that Triangle ABC is isosceles with AB = AC = 2 units.

However, the only way you could do that is if you assumed angle C is also x.

Statement1 is clearly insufficient.

However, if we can assume that angle C is x in statement 1, why can't we do the same in statement 2?

In which case we know x = y implies that all three angles are 60 degrees thus making statement 2 sufficient.

Can someone explain?
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Re: If a semicircle and triangle are pieced together to form the figure sh [#permalink]
19 Feb 2021, 10:11
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Bunuel
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