Given: HN←→ and PN←→ are perpendicular to one another.
JN and NQ¯ are horizontal segments.
HJ and PQ are vertical segments.
△HNJ and △PNQ are right triangles.
Prove: The slopes of HN←→ and PN←→ are opposite reciprocals.
Lines H N and P N intersect at point N which is in the first quadrant.
If you can show two similar triangles, you can use the proportional relationship between the side lengths of the triangles with the definitions of slope and opposite reciprocals to show that the slopes of the two perpendicular lines are opposite reciprocals.
Which pair of triangles can be proved similar to complete the proof?
A.) △HJN∼△PQN
B.) △NHJ∼△NPQ
C.) △NJH∼△NQP
D.) △HNJ∼△NPQ
The correct answer is A.) △HJN∼△PQN.
To prove that the slopes of HN←→ and PN←→ are opposite reciprocals, we need to show two similar triangles. In this case, we can show that △HJN is similar to △PQN.
Since HJ and PQ are vertical segments, they are parallel, and thus △HJN and △PQN are right triangles. We know that HN←→ and PN←→ are perpendicular to one another, so ∠JNH = 90° and ∠NQP = 90°.
Since HN←→ and PN←→ are perpendicular, the slopes of these lines are the negative reciprocals of each other.
By proving that △HJN is similar to △PQN, we can show that the sides of these triangles are proportional, which implies that the slopes of HN←→ and PN←→ are opposite reciprocals.
To complete the proof, we need to identify a pair of triangles that are similar.
Let's analyze the given information:
1. HN←→ and PN←→ are perpendicular to each other. This means that △HNJ and △PNQ are right triangles.
2. JN and NQ¯ are horizontal segments, while HJ and PQ are vertical segments.
Based on this information, we can conclude that angles HNJ and NPQ are both right angles and angles HJN and PQN are both 90 degrees.
Now let's examine the answer choices:
A.) △HJN∼△PQN: These triangles have only one angle in common, which is 90 degrees. Therefore, we cannot prove the triangles similar.
B.) △NHJ∼△NPQ: These triangles have only one angle in common, which is 90 degrees. Therefore, we cannot prove the triangles similar.
C.) △NJH∼△NQP: These triangles have only one angle in common, which is 90 degrees. Therefore, we cannot prove the triangles similar.
D.) △HNJ∼△NPQ: These triangles have two angles in common: angle HNJ and angle NPQ, which are both 90 degrees. Additionally, the third angle in both triangles is common since they complete a right angle when combined. Therefore, we can conclude that △HNJ∼△NPQ.
Since △HNJ and △NPQ are similar triangles, we can use the proportional relationship between their side lengths. The slope of a line is defined as the ratio of vertical change to horizontal change. In other words, it is the ratio of the difference in y-coordinates to the difference in x-coordinates.
Now, let's compare the two triangles:
In △HNJ, the slope of HN←→ is given by the ratio of the vertical change, NJ, to the horizontal change, HJ.
In △NPQ, the slope of PN←→ is given by the ratio of the vertical change, NQ¯, to the horizontal change, PQ.
Since △HNJ∼△NPQ, the ratio of the sides of the two triangles is equal.
Therefore, the slope of HN←→ is equal to the slope of PN←→, but with opposite sign, making them opposite reciprocals.
Hence, the correct pair of triangles to prove is D.) △HNJ∼△NPQ.