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Difficulty:
85%
(hard)
Question Stats:
43% (01:50) correct
57%
(02:06)
wrong
based on 213
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If in the semicircle above, each of the yellow angles equal to x°, what is the value of x ?
A. 48°
B. 60°
C. 72°
D. 75°
E. 78°
|
M37-34
Attachment:
Untitled8.png
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Official Solution:
If in the semicircle above, each of the yellow angles equal to \(x°\), what is the value of \(x\) ?
A. \(48°\)
B. \(60°\)
C. \(72°\)
D. \(75°\)
E. \(78°\)
Draw a mirror image of the figure around the diameter as shown below:
Consider minor arc AC. According to the The Central Angle Theorem, the measure of inscribed angle is always half the measure of the central angle (half the measure of the arc it subtends). So, if yellow angle is \(x°\), then (minor arc AC) = \(2x°\).
Say the third angle in the triangles is equal to \(y°\). The same way as above, (minor arc AC) = (minor arc AB) + (minor arc BC) = \(2y° + 2y°=4y°\).
So, we have that (minor arc AC) = \(2x = 4y\).
Finally, since the sum of the angles in a triangle is 180°, then \(x + x + y = 180\). Substitute \(y = \frac{x}{2}\) into \(2x +y = 180°\) to get \(2x +\frac{x}{2} = 180\).
Solving gives \(x = 72°\).
Answer: C
_________________
If in the semicircle above, each of the yellow angles equal to \(x°\), what is the value of \(x\) ?
A. \(48°\)
B. \(60°\)
C. \(72°\)
D. \(75°\)
E. \(78°\)
Draw a mirror image of the figure around the diameter as shown below:
Consider minor arc AC. According to the The Central Angle Theorem, the measure of inscribed angle is always half the measure of the central angle (half the measure of the arc it subtends). So, if yellow angle is \(x°\), then (minor arc AC) = \(2x°\).
Say the third angle in the triangles is equal to \(y°\). The same way as above, (minor arc AC) = (minor arc AB) + (minor arc BC) = \(2y° + 2y°=4y°\).
So, we have that (minor arc AC) = \(2x = 4y\).
Finally, since the sum of the angles in a triangle is 180°, then \(x + x + y = 180\). Substitute \(y = \frac{x}{2}\) into \(2x +y = 180°\) to get \(2x +\frac{x}{2} = 180\).
Solving gives \(x = 72°\).
Answer: C
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General Discussion
Did just a guess work
All triangles are similar, hence all the 8 angles have same value. Options have only 2 angles which are divisible by 8 (choice A=48 and C=72). As the angles are near to 90 degs. hence chose 72 degs.
All triangles are similar, hence all the 8 angles have same value. Options have only 2 angles which are divisible by 8 (choice A=48 and C=72). As the angles are near to 90 degs. hence chose 72 degs.
Posts: 1,101
We can see 5 arcs makes Total 180 degree angle at center.
Now, required angle = 2/5 *180 = 72 degree. (one arc makes angle = a/2, As angle made by joining two consecutive arcs = a)
So, I think C.
Now, required angle = 2/5 *180 = 72 degree. (one arc makes angle = a/2, As angle made by joining two consecutive arcs = a)
So, I think C.
Berdiyor
Joined: 02 Feb 2019
Last visit: 31 May 2026
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GMAT 1: 640 Q50 V25
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Hello everyone
Here is my solution
The answer is 72
C
It is easier than the rest of the two problems
Here is my solution
The answer is 72
C
It is easier than the rest of the two problems
Attachments
photo_2021-08-19_21-03-12.jpg [ 40.1 KiB | Viewed 4714 times ]
sivatx2
Joined: 23 Oct 2015
Last visit: 27 Dec 2023
Posts: 294
Location: United States (NH)
Concentration: Leadership, Technology
Schools: Wharton '25
WE:Information Technology (Non-Profit and Government)
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Schools: Wharton '25
Posts: 294
By using the inscribed angle theorem, if an inscribed angle is x, the arc it subtends is equal to 2x.
See the image attached.
As per the problem statement congruent angles is x and let make y as the angle the line makes when it touches the circumference.
Let's measure the arc BF.
Angle BAF is x. We can represent arc BF as sum of arc BD and arc DF.
arc BF = arc BD + arc DF --(1)
Based on the theorem, angle BAF, subtends arc BF.
=> arc BF = 2x. --(2)
using (1) and (2)
arc BF = arc BD + arc DC
arc BF = 2(angle DEF) + 2(angle BCD)
arc BF = 2y + 2y
arc BF = 4y
We thus have:
arc(BF) = 4y = 2x
⇒ y = x/2
Sum of three angles of a triangle = 180°
2x+y = 180°
2x + x/2 = 180°
(5/2)x = 180°
x = 72°
So the answer choice is C
See the image attached.
As per the problem statement congruent angles is x and let make y as the angle the line makes when it touches the circumference.
Let's measure the arc BF.
Angle BAF is x. We can represent arc BF as sum of arc BD and arc DF.
arc BF = arc BD + arc DF --(1)
Based on the theorem, angle BAF, subtends arc BF.
=> arc BF = 2x. --(2)
using (1) and (2)
arc BF = arc BD + arc DC
arc BF = 2(angle DEF) + 2(angle BCD)
arc BF = 2y + 2y
arc BF = 4y
We thus have:
arc(BF) = 4y = 2x
⇒ y = x/2
Sum of three angles of a triangle = 180°
2x+y = 180°
2x + x/2 = 180°
(5/2)x = 180°
x = 72°
So the answer choice is C
Attachments
InscribedAngleTheorem.png [ 64.46 KiB | Viewed 4730 times ]
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