Question

triangles pqr and lmn are graphed in the xy - plane. triangle pqr has vertices p, q, and r at (4,5), (4,7), and (6,5), respectively. triangle lmn has vertices l, m, and n at (4,5), (4,7 + k), and (6 + k,5), respectively, where k is a positive constant. if the measure of ∠q is t°, what is the measure of ∠n? (a) (90 - (t - k))° (b) (90 - (t + k))° (c) (90 - t)° (d) (90 + k)°

Explanation:

Step1: Analyze triangle PQR

Vertices of $\triangle PQR$ are $P(4,5)$, $Q(4,7)$, $R(6,5)$. Side $PQ$ is a vertical line segment (since $x$-coordinates of $P$ and $Q$ are the same, $x = 4$) with length $7 - 5=2$, and side $PR$ is a horizontal line - segment (since $y$-coordinates of $P$ and $R$ are the same, $y = 5$) with length $6 - 4 = 2$. So, $\triangle PQR$ is a right - triangle with $\angle P=90^{\circ}$, and $\angle Q+\angle R = 90^{\circ}$.

Step2: Analyze triangle LMN

Vertices of $\triangle LMN$ are $L(4,5)$, $M(4,7 + k)$, $N(6 + k,5)$. Side $LM$ is a vertical line segment (since $x$-coordinates of $L$ and $M$ are the same, $x = 4$) and side $LN$ is a horizontal line segment (since $y$-coordinates of $L$ and $N$ are the same, $y = 5$). So, $\triangle LMN$ is also a right - triangle with $\angle L = 90^{\circ}$.

Step3: Use similarity of right - triangles

Since both $\triangle PQR$ and $\triangle LMN$ are right - triangles and the slopes of their corresponding sides are the same (vertical and horizontal sides), the two triangles are similar. In right - triangle $\triangle LMN$ with $\angle L = 90^{\circ}$, and in right - triangle $\triangle PQR$ with $\angle P = 90^{\circ}$, and because of similarity, the non - right angles are equal in corresponding triangles. So, if $\angle Q=t^{\circ}$, then in $\triangle LMN$, $\angle N=(90 - t)^{\circ}$.

Answer:

C. $(90 - t)^{\circ}$