Answer :
Sure, let's carefully analyze and justify the missing step in the proof that [tex]\((PH)^2+(HT)^2=(PT)^2\)[/tex] in the context of a right triangle [tex]\(\Delta PRT\)[/tex] where [tex]\(RY\)[/tex] is the altitude.
Given a right triangle [tex]\(\Delta PRT\)[/tex] with [tex]\(R \angle PRT = 90^\circ\)[/tex] and [tex]\(RY\)[/tex] as the altitude to hypotenuse [tex]\(PT\)[/tex]:
1. [tex]\(\angle PHT\)[/tex] is congruent to [tex]\(\angle HYT\)[/tex]: Both [tex]\(\angle PHT\)[/tex] and [tex]\(\angle HYT\)[/tex] are right angles.
2. [tex]\(\angle T\)[/tex] is congruent to [tex]\(\angle T\)[/tex]: This is due to the Reflexive Property.
3. [tex]\(\Delta PHT\)[/tex] is similar to [tex]\(\Delta HYT\)[/tex]: This follows from the Angle-Angle (AA) Similarity Postulate, since they both share [tex]\(\angle T\)[/tex] and a right angle.
4. [tex]\(\frac{HT}{PT} = \frac{YT}{HT}\)[/tex]: This proportion is derived from the similarity of [tex]\(\Delta PHT\)[/tex] and [tex]\(\Delta HYT\)[/tex].
5. [tex]\((HT)^2 = (PT)(YT)\)[/tex]: This is from cross-multiplying the proportion from the previous step.
6. [tex]\(\angle PHT\)[/tex] is congruent to [tex]\(\angle PYH\)[/tex]: Both [tex]\(\angle PHT\)[/tex] and [tex]\(\angle PYH\)[/tex] are right angles which makes them congruent.
7. [tex]\(\angle P\)[/tex] is congruent to [tex]\(\angle P\)[/tex]: This is due to the Reflexive Property.
8. [tex]\(\Delta PHT\)[/tex] is similar to [tex]\(\Delta PYH\)[/tex]: This follows from the AA Similarity Postulate, since they both share [tex]\(\angle P\)[/tex] and a right angle.
9. [tex]\(\frac{PH}{PT} = \frac{PY}{PH}\)[/tex]: This proportion is derived from the similarity of [tex]\(\Delta PHT\)[/tex] and [tex]\(\Delta PYH\)[/tex].
10. [tex]\((PH)^2 = (PY)(PT)\)[/tex]: This is from cross-multiplying the proportion from the previous step.
11. [tex]\((PH)^2 + (HT)^2 = (PT)(YT) + (PT)(PY)\)[/tex]: This step combines the results from steps 5 and 10 through addition.
12. [tex]\((PH)^2 + (HT)^2 = (PT)(YT + PY)\)[/tex]: This step uses the Distributive Property to factor out [tex]\(PT\)[/tex].
The crucial missing justification is rooted in the properties of similar triangles. By knowing the similarity relationships among the smaller triangles created by the altitude [tex]\(RY\)[/tex], we establish that the segments are proportional in a manner that leads to the required sums within the triangle.
### Missing Justification:
Proportionality of triangles based on the properties of similar triangles.
This proportionality is a consequence of the fact that the altitude of a right-angled triangle to the hypotenuse creates similar triangles whose sides are proportional to the segments of the hypotenuse. This directly supports the necessary step in confirming that [tex]\((PH)^2 + (HT)^2 = (PT) \cdot (YT) + (PT) \cdot (PY)\)[/tex].
In summary, the full proof validates through similarity and proportionality of sub-triangles [tex]\(\Delta PRT\)[/tex], confirming that [tex]\((PH)^2 + (HT)^2 = (PT)^2\)[/tex].
Given a right triangle [tex]\(\Delta PRT\)[/tex] with [tex]\(R \angle PRT = 90^\circ\)[/tex] and [tex]\(RY\)[/tex] as the altitude to hypotenuse [tex]\(PT\)[/tex]:
1. [tex]\(\angle PHT\)[/tex] is congruent to [tex]\(\angle HYT\)[/tex]: Both [tex]\(\angle PHT\)[/tex] and [tex]\(\angle HYT\)[/tex] are right angles.
2. [tex]\(\angle T\)[/tex] is congruent to [tex]\(\angle T\)[/tex]: This is due to the Reflexive Property.
3. [tex]\(\Delta PHT\)[/tex] is similar to [tex]\(\Delta HYT\)[/tex]: This follows from the Angle-Angle (AA) Similarity Postulate, since they both share [tex]\(\angle T\)[/tex] and a right angle.
4. [tex]\(\frac{HT}{PT} = \frac{YT}{HT}\)[/tex]: This proportion is derived from the similarity of [tex]\(\Delta PHT\)[/tex] and [tex]\(\Delta HYT\)[/tex].
5. [tex]\((HT)^2 = (PT)(YT)\)[/tex]: This is from cross-multiplying the proportion from the previous step.
6. [tex]\(\angle PHT\)[/tex] is congruent to [tex]\(\angle PYH\)[/tex]: Both [tex]\(\angle PHT\)[/tex] and [tex]\(\angle PYH\)[/tex] are right angles which makes them congruent.
7. [tex]\(\angle P\)[/tex] is congruent to [tex]\(\angle P\)[/tex]: This is due to the Reflexive Property.
8. [tex]\(\Delta PHT\)[/tex] is similar to [tex]\(\Delta PYH\)[/tex]: This follows from the AA Similarity Postulate, since they both share [tex]\(\angle P\)[/tex] and a right angle.
9. [tex]\(\frac{PH}{PT} = \frac{PY}{PH}\)[/tex]: This proportion is derived from the similarity of [tex]\(\Delta PHT\)[/tex] and [tex]\(\Delta PYH\)[/tex].
10. [tex]\((PH)^2 = (PY)(PT)\)[/tex]: This is from cross-multiplying the proportion from the previous step.
11. [tex]\((PH)^2 + (HT)^2 = (PT)(YT) + (PT)(PY)\)[/tex]: This step combines the results from steps 5 and 10 through addition.
12. [tex]\((PH)^2 + (HT)^2 = (PT)(YT + PY)\)[/tex]: This step uses the Distributive Property to factor out [tex]\(PT\)[/tex].
The crucial missing justification is rooted in the properties of similar triangles. By knowing the similarity relationships among the smaller triangles created by the altitude [tex]\(RY\)[/tex], we establish that the segments are proportional in a manner that leads to the required sums within the triangle.
### Missing Justification:
Proportionality of triangles based on the properties of similar triangles.
This proportionality is a consequence of the fact that the altitude of a right-angled triangle to the hypotenuse creates similar triangles whose sides are proportional to the segments of the hypotenuse. This directly supports the necessary step in confirming that [tex]\((PH)^2 + (HT)^2 = (PT) \cdot (YT) + (PT) \cdot (PY)\)[/tex].
In summary, the full proof validates through similarity and proportionality of sub-triangles [tex]\(\Delta PRT\)[/tex], confirming that [tex]\((PH)^2 + (HT)^2 = (PT)^2\)[/tex].
