How to find the angle in a transformed triangle on the coordinate plane

Key takeaway: The neat part here is that the triangles are related by a shift in coordinates, so the angle change is controlled by the way the sides move. You do not need a long computation if you read the geometry correctly.

We compare the two triangles by looking at their vertices.

Triangle $PQR$

• $P=(4,5)$
• $Q=(4,7)$
• $R=(6,5)$

So $PQ$ is vertical and $PR$ is horizontal, which means $\angle P=90^\circ$.

If $\angle Q=t^\circ$, then in triangle $PQR$

$\angle R=180^\circ-90^\circ-t^\circ=90^\circ-t^\circ.$

Triangle $LMN$

• $L=(4,5)$
• $M=(4,7+k)$
• $N=(6+k,5)$

Again, $LM$ is vertical and $LN$ is horizontal, so $\angle L=90^\circ$.

The triangle is the same right-triangle pattern, just stretched outward from $L$. That means the acute angles stay aligned with the same corresponding vertices. Therefore $\angle N$ matches the angle at $R$ in the original triangle.

So

$\angle N=90^\circ-t^\circ.$

Among the choices, this is C.

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Pitfalls the pros know 👇 Do not add $k$ directly to the angle just because it appears in the coordinates. A coordinate change does not automatically mean an angle change. The angle comes from the slope relationship, and here the right-angle structure stays the same.

What if the problem changes? If the new triangle were reflected instead of stretched, the corresponding angle could swap to the other acute angle. If the vertex positions changed so that the sides were no longer horizontal and vertical, then you would need slopes or trigonometric ratios rather than symmetry alone.

Tags: right triangle, corresponding angles, coordinate plane