Perpendicular and angle bisectors questions with answers

Questions

1. Question 1On the perpendicular bisector: lengths

   In the diagram, the perpendicular bisector of $AB$ has been constructed with compasses. It meets $AB$ at $M$, and $P$ is a point on it. $AB = 8$ cm. Find the perimeter of triangle $PAB$.

   Show the worked solution

   1. Step 1: Use the property of the perpendicular bisector$PB = PA$ ($P$ is on the perpendicular bisector of $AB$)$PB = 8$ cm
   2. Step 2: Add the three sides$\text{Perimeter} = 8 + 8 + 8$$\text{Perimeter} = 24$ cm
2. Question 2On the perpendicular bisector: angles

   In the diagram, the perpendicular bisector of $AB$ has been constructed with compasses. It meets $AB$ at $M$, and $P$ is a point on it. Find $x$.

   Show the worked solution

   1. Step 1: Use the property of the perpendicular bisector$PA = PB$ ($P$ is on the perpendicular bisector of $AB$)So triangle $PAB$ is isosceles.
   2. Step 2: Find angle PBA, with a reason$\angle PBA = 57^\circ$ (base ∠s of isos. △)
   3. Step 3: PM is perpendicular to AB: use the angles in triangle PMB$\angle PMB = 90^\circ$$x = 180^\circ - 90^\circ - 57^\circ$$x = 33$ (∠ sum of △)
3. Question 3On the angle bisector

   In the diagram, the bisector of $\angle ABC$ has been constructed with compasses, and $Q$ is a point on it. $QM$ is perpendicular to $BA$ and $QN$ is perpendicular to $BC$. Find $x$.

   Show the worked solution

   1. Step 1: BQ bisects angle ABC: the two halves are equal$\angle QBN = \angle ABQ$$\angle QBN = 44^\circ$
   2. Step 2: QN is perpendicular to BC: use the angles in triangle BQN$\angle QNB = 90^\circ$$x = 180^\circ - 90^\circ - 44^\circ$$x = 46$ (∠ sum of △)
4. Question 4Which construction?

   A straight path runs between the fences $QP$ and $QR$, always the same distance from both fences. Which line does the path follow?

   Show the worked solution

   1. Step 1: What must the points be the same distance from (or a fixed distance from)?The same distance from two lines, $QP$ and $QR$.
   2. Step 2: Name the constructionPoints the same distance from the lines $QP$ and $QR$ lie on the bisector of $\angle PQR$.
5. Question 5The steps of a construction

   Ali is constructing the bisector of $\angle ABC$ with a pair of compasses and a ruler. The diagram shows the construction so far. State the next step.

   Show the worked solution

   1. Step 1: Which arcs are already drawn?The arc about $B$ has cut the arms at $X$ and $Y$.
   2. Step 2: The construction, in order1. With centre $B$, draw an arc that cuts $BA$ at $X$ and $BC$ at $Y$. (done)2. With centres $X$ and $Y$ and the same radius, draw two arcs that meet at $Z$. (next)3. Join $B$ and $Z$ with a straight line.
6. Question 6Where bisectors meet (on a grid)

   The point $P$ is the same distance from the $x$-axis and the $y$-axis, and the same distance from $A(1, 1)$ and $B(1, 9)$. $P$ is in the first quadrant. Find the coordinates of $P$.

   Show the worked solution

   1. Step 1: Same distance from the two axes: on the bisector of the angle between themIn the first quadrant that is the line $y = x$.
   2. Step 2: Points the same distance from A and B lie onSame distance from $A$ and $B$: on the perpendicular bisector of $AB$, the horizontal line through its midpoint.
   3. Step 3: Find the equation of this line$\mathstrut y = \dfrac{1 + 9}{2}$$y = 5$
   4. Step 4: On the line y = x as well$x = 5$
   5. Step 5: The point is where the two lines meet$\text{Point: } (5, 5)$