What is it's measure?
In a rectangle A B C D there are two semicircles with radius 1 7 3 cm , they just touch each other at a point G and both the semicircles also touches the point A and point C respectively. If B C measure 1 0 4 cm , then what is the measure of A B in centimetres?
Note: It is not necessary that the semicircles fully lie inside the rectangle, only there center lies on the side of the rectangle.
The answer is 676.
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2 solutions
Good, writing the big in a single line!
Let the centers of the two semi-circles be O and O ′ Join O and O ′
Draw a line from O to A B perpendicular to A B intersecting A B at J
In triangle O O ′ J
O J 2 + O ′ J 2 = O O ′ 2
1 0 4 2 + O ′ J 2 = 3 4 6 2
O ′ J = ( 3 4 6 − 1 0 4 ) ( 3 4 6 + 1 0 4 ) = 1 0 8 9 0 0 = 3 3 0
A B = O ′ A + O ′ J + O C = 1 7 3 + 3 3 0 + 1 7 3 = 6 7 6
The diagram actually helped me as I was thinking why you put O O ′ . I would suggest you to remove this line for making it harder. Thanks!
∣ A B ∣ = 1 7 3 + ( 2 × 1 7 3 ) 2 − 1 0 4 2 + 1 7 3 = 6 7 6 cm.