Constructing triangles and quadrilaterals questions with answers

Questions

1. Question 1The steps of a construction

   Ali is constructing triangle $ABC$ with $AB = 8$ cm, $\angle ABC = 65^\circ$ and $BC = 6$ cm. The diagram shows the construction so far. Which of the following is the next step?

   Show the worked solution

   1. Step 1: How many steps are done?Done: draw AB, angle at B.
   2. Step 2: The construction, in order1. Draw $AB = 8$ cm with a ruler. (done)2. At $B$, measure $\angle ABC = 65^\circ$ with a protractor and draw a ray. (done)3. With centre $B$ and radius 6 cm, draw an arc to cut the ray at $C$. (next)4. Join $AC$.
2. Question 2Can the triangle be drawn?

   State whether triangle $ABC$ can be constructed with $AB = 7$ cm, $\angle BAC = 35^\circ$ and $\angle ABC = 95^\circ$.

   Show the worked solution

   1. Step 1: Add the two angles$35 + 95 = 130$
   2. Step 2: Compare with 180°$130^\circ$ is less than $180^\circ$.
   3. Step 3: Can it be drawn? The angles of a triangle add up to 180°, so these two must add up to LESSYes: the third angle is $50^\circ$ and the rays from $A$ and $B$ cross.
3. Question 3Which data fix one triangle?

   Which set of data fixes exactly one triangle $ABC$?

   Show the worked solution

   1. Step 1: Which set gives no triangle at all?A: 3 + 4 is not more than 9, so the sides cannot meet.
   2. Step 2: Which set fixes the shape but not the size?C: three angles and no length. Triangles of any size have these angles.
   3. Step 3: Which set has an angle that is NOT between its two sides?D: $\angle BAC$ is not between $AB$ and $BC$. The arc about $B$ can cut the ray from $A$ twice: two different triangles.
   4. Step 4: So exactly one triangle comes fromB: two sides and the angle between them fix one triangle.
4. Question 4Construct a triangle and measure

   Construct triangle $ABC$ with $AB = 7$ cm, $\angle BAC = 50^\circ$ and $\angle ABC = 75^\circ$. Measure the length of $AC$, to the nearest $0.1$ cm.

   Show the worked solution

   1. Step 1: Draw the base AB with a rulerDraw $AB = 7$ cm.
   2. Step 2: At A, the angle BACAt $A$, measure $\angle BAC = 50^\circ$ and draw a ray.
   3. Step 3: At B, the angle ABCAt $B$, measure $\angle ABC = 75^\circ$ and draw a ray. The rays cross at $C$.
   4. Step 4: Measure AC with a ruler$AC = 8.3$ cm (to the nearest $0.1$ cm)
5. Question 5Construct a quadrilateral and measure

   Construct quadrilateral $ABCD$ with $AB = 7$ cm, $\angle DAB = 105^\circ$, $AD = 5$ cm, $\angle ABC = 110^\circ$ and $BC = 6$ cm. Measure the length of $CD$, to the nearest $0.1$ cm.

   Show the worked solution

   1. Step 1: Draw the base AB with a rulerDraw $AB = 7$ cm.
   2. Step 2: At A: the angle DAB, then DAt $A$, measure $\angle DAB = 105^\circ$, draw a ray and mark $D$ on it with $AD = 5$ cm.
   3. Step 3: At B: the angle ABC, then CAt $B$, measure $\angle ABC = 110^\circ$, draw a ray and mark $C$ on it with $BC = 6$ cm.
   4. Step 4: Join CD and measure it$CD = 10.4$ cm (to the nearest $0.1$ cm)