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The diagram below shows the parallelogram PARK - Junior Cycle Mathematics - Question 11 - 2015
Question 11
The diagram below shows the parallelogram PARK. (a) Construct the bisector of ∠KPA on the diagram above, using only a compass and straight edge. Show your construct... show full transcript
Worked Solution & Example Answer:The diagram below shows the parallelogram PARK - Junior Cycle Mathematics - Question 11 - 2015
Step 1
Construct the bisector of ∠KPA
Answer
- Place the compass point at vertex P of angle ∠KPA.
- Draw an arc that intersects both rays PK and PA. Label the points of intersection as X and Y.
- Without changing the width of the compass, place the compass point on X and draw an arc above ∠KPA. Label this intersection as A1.
- Next, place the compass point on Y and draw another arc that intersects the previous arc (A1). Label this intersection as A2.
- Using a straight edge, draw a line from point P to the intersection A2. This line PA2 is the bisector of angle ∠KPA.
Step 2
Prove that all four sides of the parallelogram are equal in length.
Answer
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|HO| = |EP| and |HE| = |OP| ..........Opposite sides of parallelogram
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|OH| = |OP| .............................Isosceles triangle PHO ⇒ |EH| = |OP| = |HO| = |EP|
or
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|∠PHO| = |∠EPH| ....................Alternate angles
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|∠OPH| = |∠EHP| ....................Alternate angles
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|HP| = |HP| ...........................Common side
∴ ΔHOP ≅ ΔHEP ......................A.S.A ∴ |OP| = |EH| and |HO| = |EP| .......... Corresponding sides
- |HO| = |OP| ............................Isosceles triangle PHO
- |HE| = |OP| = |HO| = |EP|