During resurfacing work, a car park layout is marked out as an irregular pentagon, shown in the diagram, with interior angles \((2x-10)^{\circ}\), \((3x+5)^...
Question 1 Report
During resurfacing work, a car park layout is marked out as an irregular pentagon, shown in the diagram, with interior angles \((2x-10)^{\circ}\), \((3x+5)^{\circ}\), \((x+15)^{\circ}\), \((4x-20)^{\circ}\) and \((2x+10)^{\circ}\).
- Form an equation in \(x\) and solve it. (2)
- Find the size of the largest interior angle. (1)
- State, with a reason, whether the pentagon could be regular. (1)
Answer Details
This question forms and solves a linear equation from the interior angle sum of a pentagon expressed algebraically, then uses the solution to find each angle and assess whether the pentagon could be regular.
The interior angles of a pentagon sum to \((5-2)\times180^\circ=540^\circ\). Adding the five given expressions:
\[(2x-10)+(3x+5)+(x+15)+(4x-20)+(2x+10) = 12x = 540\]Dividing by \(12\):
\[x = 45\][2 marks]
Substituting \(x=45\) into each expression gives angles of \(80^\circ, 140^\circ, 60^\circ, 160^\circ, 100^\circ\). The largest is \(160^\circ\) [1 mark].
Since the five angles \(80^\circ, 140^\circ, 60^\circ, 160^\circ, 100^\circ\) are not all equal, the pentagon could not be regular [1 mark].
Checking: \(80+140+60+160+100=540^\circ\), confirming the angles found are consistent with the pentagon's angle sum, even though they are not all the same size.
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