{"id":57700,"date":"2025-08-27T21:33:26","date_gmt":"2025-08-27T16:03:26","guid":{"rendered":"https:\/\/e-gmat.com\/blogs\/?p=57700"},"modified":"2025-09-05T00:00:36","modified_gmt":"2025-09-04T18:30:36","slug":"right-triangle-pqr-is-to-be-constructed-in-the-xy-plane-so","status":"publish","type":"post","link":"https:\/\/e-gmat.com\/blogs\/right-triangle-pqr-is-to-be-constructed-in-the-xy-plane-so\/","title":{"rendered":"Right triangle PQR is to be constructed in the xy-plane so that the right angle is at P&#8230;."},"content":{"rendered":"<span class=\"rt-reading-time\" style=\"display: block;\"><span class=\"rt-label rt-prefix\">A <\/span> <span class=\"rt-time\">4<\/span> <span class=\"rt-label rt-postfix\">min read <\/span><\/span>\n<p>Right triangle PQR is to be constructed in the xy-plane so that the right angle is at P and PR is parallel to the x-axis. The x and Y coordinates of P, Q and R are to be integers that satisfy the inequalities -4 \u2264 x \u2264 5 and 6 \u2264 y \u2264 16. How many different triangles with these properties could be constructed?<\/p>\n\n\n\n<p><strong>Options:<\/strong><\/p>\n\n\n\n<ul><li>A. 110<\/li><li>B. 1,100<\/li><li>C. 9,900<\/li><li>D. 10,000<\/li><li>E. 12,100<\/li><\/ul>\n\n\n\n<h2 id=\"h-2-solution-section\">2. Solution Section:<\/h2>\n\n\n\n<p><strong>Solution:<\/strong><\/p>\n\n\n\n<ul><li>Translate the problem requirements: We need to count right triangles PQR where P is the right angle vertex, PR is horizontal (parallel to x-axis), and all vertices have integer coordinates within specified bounds (-4 \u2264 x \u2264 5, 6 \u2264 y \u2264 16).<\/li><li>Identify geometric constraints from the right angle condition: Since angle P is 90\u00b0 and PR is horizontal, side PQ must be vertical (perpendicular to PR), which restricts how we can place the three points.<\/li><li>Count valid positions for each vertex systematically: Determine how many coordinate positions are available for P, then for each P position, count valid positions for Q and R that satisfy the geometric constraints.<\/li><li>Apply the multiplication principle: Calculate total triangles by multiplying the number of choices for each vertex, ensuring we account for the independence of choices.<\/li><\/ul>\n\n\n\n<h3 id=\"h-execution-of-strategic-approach\">Execution of Strategic Approach<\/h3>\n\n\n\n<ol><li>Translate the problem requirements Let&#8217;s break down what we need to understand from the problem statement:<\/li><\/ol>\n\n\n\n<p>What is a right triangle with these properties?<\/p>\n\n\n\n<ul><li>We have three points P, Q, and R that form a triangle<\/li><li>The right angle (90\u00b0) is at point P<\/li><li>Side PR must be parallel to the x-axis (horizontal)<\/li><li>All three points must have integer coordinates<\/li><li>All coordinates must satisfy: -4 \u2264 x \u2264 5 and 6 \u2264 y \u2264 16<\/li><\/ul>\n\n\n\n<p>What does &#8220;different triangles&#8221; mean? Two triangles are different if they have at least one vertex at a different location. So we&#8217;re counting all possible ways to place three points P, Q, and R that satisfy our conditions.<\/p>\n\n\n\n<p>Available coordinate grid:<\/p>\n\n\n\n<ul><li>x-values: -4, -3, -2, -1, 0, 1, 2, 3, 4, 5 (that&#8217;s 10 possible values)<\/li><li>y-values: 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16 (that&#8217;s 11 possible values)<\/li><\/ul>\n\n\n\n<p>Process Skill: TRANSLATE &#8211; Converting the geometric language into clear coordinate constraints<\/p>\n\n\n\n<ol start=\"2\"><li>Identify geometric constraints from the right angle condition What does &#8220;right angle at P&#8221; mean geometrically? If angle P is 90\u00b0, then the two sides meeting at P (which are PQ and PR) must be perpendicular to each other.<\/li><\/ol>\n\n\n\n<p>What does &#8220;PR parallel to x-axis&#8221; tell us? If PR is horizontal (parallel to x-axis), this means:<\/p>\n\n\n\n<ul><li>Points P and R have the same y-coordinate<\/li><li>Points P and R have different x-coordinates<\/li><li>So if P is at (x\u2081, y\u2081), then R must be at (x\u2082, y\u2081) where x\u2082 \u2260 x\u2081<\/li><\/ul>\n\n\n\n<p>What about point Q? Since PQ must be perpendicular to the horizontal line PR, PQ must be vertical. This means:<\/p>\n\n\n\n<ul><li>Points P and Q have the same x-coordinate<\/li><li>Points P and Q have different y-coordinates<\/li><li>So if P is at (x\u2081, y\u2081), then Q must be at (x\u2081, y\u2082) where y\u2082 \u2260 y\u2081<\/li><\/ul>\n\n\n\n<p>Process Skill: INFER &#8211; Drawing the non-obvious conclusion that PQ must be vertical<\/p>\n\n\n\n<ol start=\"3\"><li>Count valid positions for each vertex systematically How many ways can we choose point P? Point P can be anywhere on our grid, so:<\/li><\/ol>\n\n\n\n<ul><li>P has 10 choices for x-coordinate \u00d7 11 choices for y-coordinate = 110 possible positions<\/li><\/ul>\n\n\n\n<p>For each choice of P, how many ways can we choose R? Once P is chosen at position (x\u2081, y\u2081), point R must be at (x\u2082, y\u2081) where:<\/p>\n\n\n\n<ul><li>R has the same y-coordinate as P (so y-coordinate is fixed)<\/li><li>R has a different x-coordinate than P<\/li><li>Since there are 10 possible x-values total, and we exclude P&#8217;s x-coordinate, R has 9 possible positions<\/li><\/ul>\n\n\n\n<p>For each choice of P, how many ways can we choose Q? Once P is chosen at position (x\u2081, y\u2081), point Q must be at (x\u2081, y\u2082) where:<\/p>\n\n\n\n<ul><li>Q has the same x-coordinate as P (so x-coordinate is fixed)<\/li><li>Q has a different y-coordinate than P<\/li><li>Since there are 11 possible y-values total, and we exclude P&#8217;s y-coordinate, Q has 10 possible positions<\/li><\/ul>\n\n\n\n<p>Process Skill: APPLY CONSTRAINTS &#8211; Systematically accounting for the geometric restrictions<\/p>\n\n\n\n<ol start=\"4\"><li>Apply the multiplication principle How do we count the total number of triangles? Since the choices are independent:<\/li><\/ol>\n\n\n\n<ul><li>First, choose P: 110 ways<\/li><li>Then, choose R: 9 ways (for each choice of P)<\/li><li>Then, choose Q: 10 ways (for each choice of P)<\/li><\/ul>\n\n\n\n<p>Total number of triangles: 110 \u00d7 9 \u00d7 10 = 110 \u00d7 90 = 9,900<\/p>\n\n\n\n<p>Let&#8217;s verify this makes sense:<\/p>\n\n\n\n<ul><li>We have 10 \u00d7 11 = 110 positions for P \u2713<\/li><li>For each P, we have 9 positions for R (same row, different column) \u2713<\/li><li>For each P, we have 10 positions for Q (same column, different row) \u2713<\/li><li>Total: 110 \u00d7 9 \u00d7 10 = 9,900 \u2713<\/li><\/ul>\n\n\n\n<ol start=\"5\"><li>Final Answer The total number of different right triangles that can be constructed is 9,900.<\/li><\/ol>\n\n\n\n<p>This matches answer choice (C) 9,900.<\/p>\n\n\n\n<h3 id=\"h-common-faltering-points\">Common Faltering Points<\/h3>\n\n\n\n<p>Errors while devising the approach<\/p>\n\n\n\n<ol><li>Misunderstanding what &#8220;right angle at P&#8221; means geometrically Many students struggle to translate &#8220;right angle at P&#8221; into the constraint that sides PQ and PR must be perpendicular. They might think this just means P is a corner of the triangle, without realizing it creates specific geometric restrictions on how Q and R can be positioned relative to P.<\/li><li>Missing the perpendicularity constraint Even when students understand that PR is parallel to the x-axis (horizontal), they often fail to make the crucial connection that PQ must therefore be perpendicular to PR, meaning PQ must be vertical. This leads them to allow Q to be placed anywhere on the grid rather than restricting it to the same x-coordinate as P.<\/li><li>Not recognizing this as a systematic counting problem Students may attempt to visualize or draw specific triangles rather than recognizing this as a problem where they need to count all possible valid combinations of three points. This leads to inefficient approaches or incomplete counting.<\/li><\/ol>\n\n\n\n<p>Errors while executing the approach<\/p>\n\n\n\n<ol><li>Incorrectly counting available positions for R and Q Students often forget to exclude P&#8217;s position when counting options for R and Q. For example, they might count 10 positions for R instead of 9 (forgetting that R cannot occupy the same position as P), or 11 positions for Q instead of 10 (forgetting that Q cannot have the same y-coordinate as P).<\/li><li>Arithmetic errors in the final multiplication When calculating 110 \u00d7 9 \u00d7 10, students may make computational mistakes, especially in intermediate steps like 9 \u00d7 10 = 90, leading to incorrect final answers.<\/li><li>Confusing the coordinate ranges Students may miscalculate the total number of available x-values (thinking there are 9 instead of 10) or y-values (thinking there are 10 instead of 11) by incorrectly handling the inclusive ranges -4 \u2264 x \u2264 5 and 6 \u2264 y \u2264 16.<\/li><\/ol>\n\n\n\n<p>Errors while selecting the answer<\/p>\n\n\n\n<p>No likely faltering points &#8211; once students have calculated 9,900 correctly, the answer choice (C) is clearly identifiable and matches exactly.<\/p>\n","protected":false},"excerpt":{"rendered":"<p>Right triangle PQR is to be constructed in the xy-plane so that the right angle is at P and PR is parallel to the x-axis. The x and Y coordinates of P, Q and R are to be integers that satisfy the inequalities -4 \u2264 x \u2264 5 and 6 \u2264 y \u2264 16. How [&hellip;]<\/p>\n","protected":false},"author":102457,"featured_media":0,"comment_status":"closed","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"_et_pb_use_builder":"off","_et_pb_old_content":"","_et_gb_content_width":"","ub_ctt_via":""},"categories":[44,100],"tags":[],"featured_image_src":null,"author_info":{"display_name":"Kashish Garg","author_link":"https:\/\/e-gmat.com\/blogs\/author\/kashish\/"},"yoast_head":"<!-- This site is optimized with the Yoast SEO Premium plugin v17.1.1 (Yoast SEO v17.1) - https:\/\/yoast.com\/wordpress\/plugins\/seo\/ -->\n<title>Right triangle PQR is to be constructed in the xy-plane so that the right angle is at P....<\/title>\n<meta name=\"description\" content=\"Right triangle PQR is to be constructed in the xy-plane so that the right angle is at P and PR is parallel to the x-axis\" \/>\n<meta name=\"robots\" content=\"index, follow, max-snippet:-1, max-image-preview:large, max-video-preview:-1\" \/>\n<link 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