In the diagram below, $\angle MNP = \angle PRQ$. (i) Prove that $ \triangle MNP $ and $ \triangle QRP $ are similar. (ii) Is $ NM $ parallel to $ QR $? Give a reason for your answer. Given $ |M| = 6, |P| = 4, |Q| = 9, $ and $ |R| = 10, $ find: (iii) Find $ |QR| $. (iv) Find $ |QM| $.

Junior Cycle Mathematics Geometry: In the diagram below, $\angle MN

In the diagram below, $\angle MNP = \angle PRQ$ - Junior Cycle Mathematics - Question i - 2014

Question i

$In-the-diagram-below,-$\angle-MNP-=-\angle-PRQ$-Junior Cycle Mathematics-Question i-2014.png$

In the diagram below, $\angle MNP = \angle PRQ$. (i) Prove that $ \triangle MNP $ and $ \triangle QRP $ are similar. (ii) Is $ NM $ parallel to $ QR $?... show full transcript

Worked Solution & Example Answer:In the diagram below, $\angle MNP = \angle PRQ$ - Junior Cycle Mathematics - Question i - 2014

Step 1

Prove that $ \triangle MNP $ and $ \triangle QRP $ are similar.

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Answer

To prove that the triangles are similar, we can use the AA (Angle-Angle) similarity criterion.

Given that ( \angle MNP = \angle PRQ ) (given).
Since ( \angle MNP ) and ( \angle PRQ ) are vertically opposite angles, they are equal.
Therefore, ( \angle NMP ) is equal to ( \angle RQP ) (third angles).

Thus, by AA, ( \triangle MNP ) is similar to ( \triangle QRP ).

Step 2

Is $ NM $ parallel to $ QR $? Give a reason for your answer.

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Answer

Yes, ( NM ) is parallel to ( QR ) because ( \angle MNP = \angle PRQ ) or alternate angles are equal.

Step 3

Find $ |QR| $.

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Answer

By similar triangles ( \triangle MNP ) and ( \triangle QRP ): [ \frac{|M|}{|N|} = \frac{|QR|}{|P|} ] [ \frac{6}{4} = \frac{|QR|}{10} ] Cross-multiplying gives ( 6 \cdot 10 = 4 \cdot |QR| ). Thus, ( |QR| = \frac{60}{4} = 15 ).

Step 4

Find $ |QM| $.

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Answer

By similar triangles ( \triangle MNP ) and ( \triangle QRP ):
Using the length of the sides: [ |QM| = \frac{|M|}{|QR|} \times |Q| ] Substituting the values: [ |QM| = \frac{6}{15} \times 9 = 3.6 ] Alternatively, you can also find ( |QM| ) using:
[ |QM| = |Q| + |M| = 9 + 3.6 = 12.6 ].

In the diagram below, $\angle MNP = \angle PRQ$ - Junior Cycle Mathematics - Question i - 2014

Worked Solution & Example Answer:In the diagram below, $\angle MNP = \angle PRQ$ - Junior Cycle Mathematics - Question i - 2014

Prove that \( \triangle MNP \) and \( \triangle QRP \) are similar.

Is \( NM \) parallel to \( QR \)? Give a reason for your answer.

Find \( |QR| \).

Find \( |QM| \).

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