In the figure shown, the triangle is inscribed in the semicircle. If

Question

https://gmatclub.com/forum/download/file.php?id=28115 In the figure shown, the triangle is inscribed in the semicircle. If the length of line segment AB is 8 and the length of line segment BC is 6, what is the length of arc ABC? (A) 15π (B) 12π (C) 10π (D) 7π (E) 5π 2015-10-19_0005.png

ScottTargetTestPrep · Accepted Answer

https://i.imgur.com/QHjgelz.png

We are given that triangle ABC is inscribed in the semicircle. Since a triangle inscribed in a semicircle is always a right triangle, triangle ABC is a right triangle. Therefore, AC is both the hypotenuse of triangle ABC and the diameter of the semicircle. We are also given that side AB = 8 and that side BC = 6. With that information, we can conclude that triangle ABC is a 6-8-10 right triangle, in which side AC = 10.

We need to determine the length of arc ABC, or, in other words, the length of half of the circumference of the circle. Since the diameter is 10, the radius is 5. We use the circumference formula C = 2∏r to obtain 10∏ as the circumference of the entire circle. The length of arc ABC is half of the circumference; therefore, its value is 5∏.

Answer: E

Bunuel · Answer

A right triangle inscribed in a circle must have its hypotenuse as the diameter of the circle. The reverse is also true: if the diameter of the circle is also the triangle’s hypotenuse, then that triangle is a right triangle.

So as ABC is inscribed in semicircle then ABC is a right triangle and AC=diameter=hypotenuse. Also note that the length of arc ABC is half of the circumference.

AC=\sqrt{AB^2+BC^2}=\sqrt{64+36}=10 ;

radius=\frac{diameter}{2}=\frac{AC}{2}=5 ;

circumference=2\pi{r}=10\pi ;

arc_{ABC}=\frac{circumference}{2}=5\pi .

Answer: E.

camlan1990 · Answer

the calculated value = half of the perimeter of circle O = pi*R

The length of AC = 10 => R = 5 => the calculated value = Pi * 5

Ans E

mvictor · Answer

without any calculations, we can see that AC is the diagonal of a circle. Knowing that AB=8 and AC=6 we can spot the Pythagorean triplet 3-4-5, and can deduce that AC, the diagonal is 10. Circumference of the circle must then be 10pi. Since we have only a semicircle, the length of the arc then must be 1/2*10pi = 5pi. Answer choice E.

glt13 · Answer

I am looking for the rule that states that any angle like <B in this problem will always be right angle, can't find it in my pocket reference. Theoretical explanation from anyone? Thanks

Bunuel · Answer

A right triangle's hypotenuse is a diameter of its circumcircle (circumscribed circle).

The reverse is also true: if one of the sides of an inscribed triangle is a diameter of the circle, then the triangle is a right angled (right angel being the angle opposite the diameter/hypotenuse).

mcelroytutoring · Answer

Attached is a visual that should help.

Brego7 · Answer

Bunuel 
 ScottTargetTestPrep

Guys,

If a triangle is inscribed in a semi circle then it is always right triangle? If a triangle is inscribed in a circle, is it also always a right triangle?

Bunuel · Answer

A  right  triangle inscribed in a circle must have its hypotenuse as the diameter of the circle.  The reverse is also true: if the diameter of the circle is also the triangle's side, then that triangle is a right triangle.

If the diameter of the circle is NOT inscribed triangle's side, then that triangle is NOT a right triangle.

Brego7 · Answer

Great response  Bunuel . I was wondering if inscribed right triangle has circle's diameter for one of its sides, will the triangle have to be 30-60-90 or 45-45-90 triangle? Or right triangle can have any degree values for 2 other sides?

Thanks!

Bunuel · Answer

Other two angles can take any degree measures from 0 to 90, not inclusive.

EMPOWERgmatRichC · Answer

Hi All,

In this prompt, we are told that a triangle is INSCRIBED in a semi-circle (which means that all 3 points of the triangle are ON the circumference of the half-circle) – and we are given the lengths of the two shorter sides of the triangle (6 and 8). We’re asked for the ARC LENGTH of the semi-circle.

To answer this question, we’ll clearly need the radius of the semi-circle. There is also one specific math rule that we’ll need to know for this question: when a triangle is inscribed in a circle – AND one of the sides is the DIAMETER of the circle – then we have a RIGHT TRIANGLE.

Since triangle ABC is a right triangle and the two ‘legs’ are 6 and 8, we have a 3/4/5 right triangle that has been ‘doubled’ (meaning that the sides are 6, 8 and 10). By extension, the diameter of the semi-circle is 10 and the radius is 5.

Circumference of a Circle is based on the formula:

C = (2)(pi)(Radius)

A semi-circle is half of a circle, so once we have the full circumference, we can divide that by 2 to find the arc length here:

(2)(pi)(5) / 2 = 5pi

Final Answer:  E

GMAT Assassins aren’t born, they’re made,
Rich

GMATinsight · Answer

Answer: Option E 
  https://www.youtube.com/watch?v=P2uHyZA17CY

avigutman · Answer

Video solution from Quant Reasoning: Subscribe for more: https://www.youtube.com/QuantReasoning? ... irmation=1 https://www.youtube.com/watch?v=jRuqHqnIQg0

bumpbot · Answer

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In the figure shown, the triangle is inscribed in the semicircle. If

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