Question: Q1. The diagram shows two circles touching a straight line at $X$ and $Y$ and a common tangent line through their tops. a) If $OX$, and $OP$ are $6$ cm and $7$ cm respectively, find the area of $OXY$. b) (i) Show that the tangent line is $8$ cm and $6$ cm. (ii) If $OQ = 10$ cm, find the length of the line segment.

1. **Problem Statement:** We have two circles touching a straight line at points $X$ and $Y$ respectively, and a common tangent line passing through their top points. Given $OX = 6$ cm and $OP = 7$ cm, we need to find the area of triangle $OXY$. Then, we need to show the tangent line lengths are $8$ cm and $6$ cm, and finally, given $OQ = 10$ cm, find the length of the line segment $PQ$. --- 2. **Step a) Find area of triangle $OXY$** - The points $O$, $X$, and $Y$ form a triangle. - $OX = 6$ cm is one side. - $OP = 7$ cm is given, but we need to understand the triangle's dimensions. Assuming $OXY$ is a right triangle with $OX$ as base and $XY$ as height. - Since the circles touch the line at $X$ and $Y$, and the tangent line passes through the tops, the height $XY$ can be found using the difference in radii or given lengths. - Without explicit coordinates, we use the tangent lengths from part b) to find $XY$. --- 3. **Step b) (i) Show tangent line lengths are $8$ cm and $6$ cm** - The tangent lengths from the points of tangency to the tangent line are given or to be shown. - Using Pythagoras theorem in the right triangles formed by the radii and tangent lines: For the circle with radius $r_1$ and tangent length $t_1$: $$ t_1 = \sqrt{OP^2 - OX^2} = \sqrt{7^2 - 6^2} = \sqrt{49 - 36} = \sqrt{13} \approx 3.6 \text{ cm} $$ This does not match $8$ cm or $6$ cm, so likely the tangent lengths correspond to other segments. Assuming the tangent line segments are $8$ cm and $6$ cm as given, we accept these lengths. --- 4. **Step b) (ii) Find length of line segment $PQ$ given $OQ = 10$ cm** - Using the right triangle formed by $O$, $P$, and $Q$: $$ PQ = \sqrt{OQ^2 - OP^2} = \sqrt{10^2 - 7^2} = \sqrt{100 - 49} = \sqrt{51} \approx 7.14 \text{ cm} $$ --- 5. **Final answers:** - Area of triangle $OXY$ can be approximated if $XY$ is known. - Tangent line lengths are $8$ cm and $6$ cm as given. - Length of segment $PQ \approx 7.14$ cm. --- **Note:** Due to incomplete data and diagram, assumptions were made based on typical tangent and circle properties.