A farmer fences a triangular paddock \(ABC\) so that the two fences \(AB\) and \(AC\) are equal in length. The apex angle at \(A\) is \((4x)^\circ\) and eac...
Question 1 Report
A farmer fences a triangular paddock \(ABC\) so that the two fences \(AB\) and \(AC\) are equal in length. The apex angle at \(A\) is \((4x)^\circ\) and each base angle is \((x + 15)^\circ\), as shown.
- Form and solve an equation to find \(x\). (2)
- State the size of the apex angle at \(A\). (1)
Answer Details
This question uses the angle sum of a triangle, together with the fact that two base angles of an isosceles triangle are equal, to form and solve an equation.
Since \(AB = AC\), the base angles at \(B\) and \(C\) are equal, each \((x+15)^\circ\). The three angles of triangle \(ABC\) sum to \(180^\circ\):
\[4x + (x+15) + (x+15) = 180\] \[6x + 30 = 180\] \[6x = 150\] \[x = 25\][2 marks]
Substituting \(x=25\) into the apex angle expression:
\[4(25) = 100^\circ\][1 mark]
Recognising that the two fences of equal length force the two base angles to be equal is the key step; without that fact there would be three unknown angles and only one equation.
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