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In two congruent squares ABCD and AEFG shown above, points C, E, H and G are collinear. What is the measure of yellow angle ?
A. \(95°\)
B. \(100°\)
C. \(105°\)
D. \(110°\)
E. \(115°\)
23 Nov 2021, 04:08
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Official Solution:
In two congruent squares ABCD and AEFG shown above, points C, E, H and G are collinear. What is the measure of yellow angle ?
A. \(95°\)
B. \(100°\)
C. \(105°\)
D. \(110°\)
E. \(115°\)
Say the length of the side of the square is \(x\). Consider triangle AOC below:
AC is the diagonal of the square and thus will equal to \(x\sqrt{2}\)
AO is half of the diagonal of the square and thus will equal to \(\frac{x\sqrt{2}}{2}\)
So, the ratio of the hypotenuse to one of the legs is 2:1. This means that we have a 30°-60°-90° triangle. In such triangles, the sides are in the ratio \(1:√3:2\). From this it follows that the angle opposite the smallest side (AO) would be 30°. This makes angle HCD equal to \(45° - 30°=15°\), angle DHC equal to \(90° - 15°=75°\) and finally angle OHD equal to \(180° - 75°=105°\)
Answer: C
In two congruent squares ABCD and AEFG shown above, points C, E, H and G are collinear. What is the measure of yellow angle ?
A. \(95°\)
B. \(100°\)
C. \(105°\)
D. \(110°\)
E. \(115°\)
Say the length of the side of the square is \(x\). Consider triangle AOC below:
AC is the diagonal of the square and thus will equal to \(x\sqrt{2}\)
AO is half of the diagonal of the square and thus will equal to \(\frac{x\sqrt{2}}{2}\)
So, the ratio of the hypotenuse to one of the legs is 2:1. This means that we have a 30°-60°-90° triangle. In such triangles, the sides are in the ratio \(1:√3:2\). From this it follows that the angle opposite the smallest side (AO) would be 30°. This makes angle HCD equal to \(45° - 30°=15°\), angle DHC equal to \(90° - 15°=75°\) and finally angle OHD equal to \(180° - 75°=105°\)
Answer: C
