Geometry

To find the value of angle 'x' in the quadrilateral, we will use the property that the sum of the interior angles of a quadrilateral is .
Let's assume the given angles in the quadrilateral are , , , and . According to the property:
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The figure provides specific angles, which we will add together and subtract from to find the value of .
If the provided angles are , , and , then:
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Solving for , we subtract from both sides:
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Therefore, the value of angle 'x' is .

Let the length of the chord be . The distance from the center to the chord is given as 12 cm, and the radius of the circle is 13 cm.
Using the right triangle formed by the radius, the distance from the center to the chord (perpendicular), and half the chord length, we apply Pythagoras theorem:
Therefore, the length of the chord .

Given two circles intersecting at points A and B. PQ and RS are common tangents touching the circles. Points X and Y are where the lines through A and B meet the tangents PQ and RS respectively.
From the properties of intersecting circles and their common tangents, the length can be found by: However, from the figure and properties of tangents, the correct relation is: