Six cone markers are placed around a hexagonal running track layout for school sports day. The interior angles of the hexagon formed by the markers are \(2x...
Question 1 Report
Six cone markers are placed around a hexagonal running track layout for school sports day. The interior angles of the hexagon formed by the markers are \(2x^{\circ}\), \(3x^{\circ}\), \((x+10)^{\circ}\), \((2x-8)^{\circ}\), \((x+16)^{\circ}\) and \((3x-6)^{\circ}\), shown in the diagram.
- Write down an equation in \(x\) using the angle sum of a hexagon. (1)
- Solve your equation to find the value of \(x\). (2)
- Work out the size of the largest interior angle. (2)
- State, with a reason, whether the hexagonal layout is convex. (2)
Answer Details
The interior angles of any hexagon sum to \((6-2)\times180=720^{\circ}\), which turns the picture into an equation in \(x\).
(a) Adding all six given angles and setting the total equal to \(720\):
\[2x+3x+(x+10)+(2x-8)+(x+16)+(3x-6)=720\] [1 mark](b) Collecting the \(x\) terms (\(2+3+1+2+1+3=12\)) and the constants (\(10-8+16-6=12\)) gives
\[12x+12=720\]so \(12x=708\) and \(x=59\). [2 marks]
(c) Substituting \(x=59\) into each expression: \(2(59)=118^{\circ}\), \(3(59)=177^{\circ}\), \(59+10=69^{\circ}\), \(2(59)-8=110^{\circ}\), \(59+16=75^{\circ}\), \(3(59)-6=171^{\circ}\). These sum to \(118+177+69+110+75+171=720^{\circ}\), confirming the value of \(x\). The largest interior angle is \(177^{\circ}\). [2 marks]
(d) A polygon is convex exactly when every interior angle is less than \(180^{\circ}\) (no vertex "caves inwards"). Checking the largest angle found in part (c), \(177^{\circ} \lt 180^{\circ}\), and all the other angles (\(118^{\circ}, 69^{\circ}, 110^{\circ}, 75^{\circ}, 171^{\circ}\)) are smaller still, so every interior angle is below \(180^{\circ}\). The layout is therefore convex. [2 marks]
Because it is only the largest angle that could possibly break the convexity test, checking it (rather than all six individually) is enough to justify the conclusion once its value is known.
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