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PQR LMN Triangles and are graphed in the PQR xy-plane. Triangle has vertices P, Q, and R at (4,5) (4,7) (6,5) LMN,, and, respectively. Triangle (4,5) (4,7+k) has vertices L, M, and N at,, (6 +k,5) and, respectively, where k is a positive ∠Q t° constant. If the measure of is, what is the ∠N measure of? (90−(t−k))°
Correct answer: C (parsed from the explanation — this source omits an answer key for this item)
Explanation
= 4,5 = 4,7 Choice C is correct. Since P ^ h and Q ^ h, side PQ is parallel to the y-axis = 4,5 = 6,5 and has a length of 2. Since P ^ h and R ^ h, side PR is parallel to the x-axis and has a length of 2. Therefore, triangle PQR is a right isosceles triangle, where + 90c + + 45c P has measure and Q and R each have measure . It follows that if + c =45 = 4,5 = 4,7+ the measure of Q is t , then t . Since L ^ h and M ^ kh, side LM is + = 4,5 = 6+ ,5 parallel to the y-axis and has a length of k 2. Since L ^ h and N ^ k h, + side LN is parallel to the x-axis and has a length of k 2. Therefore, triangle LMN + 90c + + is a right isosceles triangle, where L has measure and M and N each 45c 90- c 45c have measure . Of the given choices, only ^ th is equal to , so the + 90- c measure of N is ^ th. Choice A is incorrect and may result from conceptual or calculation errors. Choice B is incorrect and may result from conceptual or calculation errors. Choice D is incorrect and may result from conceptual or calculation errors.
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