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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. \(78°\)
23 Nov 2021, 06:00
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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
