{"@context":"https://schema.org/","@type":"Quiz","name":"

The length of a rectangle's diagonal is $5\\sqrt{17}$, and the length of the rectangle's shorter side is 5. What is the length of the rectangle's longer side?

","hasPart":{"@type":"Question","typicalAgeRange":"12-22","educationalAlignment":[{"@type":"AlignmentObject","alignmentType":"educationalSubject","targetName":"SAT"},{"@type":"AlignmentObject","alignmentType":"educationalLevel","targetName":"SAT","educationalFramework":"Common Core","targetUrl":"https://classnotes.xyz/mcqs/sat-math-geometry-and-trigonometry-multiple-choice-question-49"}],"about":{"@type":"Thing","name":"Math, Geometry and Trigonometry, Duplicate"},"educationalLevel":"intermediate","eduQuestionType":"Multiple choice","learningResourceType":"Practice problem","assesses":["Math","Geometry and Trigonometry","Duplicate"],"name":"

The length of a rectangle's diagonal is $5\\sqrt{17}$, and the length of the rectangle's shorter side is 5. What is the length of the rectangle's longer side?

","comment":{"@type":"Comment","text":"

Choice B is correct. A rectangle's diagonal divides a rectangle into two congruent right triangles, where the diagonal is the hypotenuse of both triangles. It's given that the length of the diagonal is $5\\sqrt{17}$ and the length of the rectangle's shorter side is 5. Therefore, each of the two right triangles formed by the rectangle's diagonal has a hypotenuse with length $5\\sqrt{17}$, and a shorter leg with length 5. To calculate the length of the longer leg of each right triangle, the Pythagorean theorem, $a^{2}+b^{2}=c^{2}$, can be used, where $a$ and $b$ are the lengths of the legs and $c$ is the length of the hypotenuse of the triangle. Substituting 5 for $a$ and $5\\sqrt{17}$ for $c$ in the equation $a^{2}+b^{2}=c^{2}$ yields $5^{2}+b^{2}=\\left(5\\sqrt{17}\\right)^{2}$, which is equivalent to $25+b^{2}=25(17)$, or $25+b^{2}=425$. Subtracting 25 from each side of this equation yields $b^{2}=400$. Taking the positive square root of each side of this equation yields $b=20$. Therefore, the length of the longer leg of each right triangle formed by the diagonal of the rectangle is 20. It follows that the length of the rectangle's longer side is 20.

Choice A is incorrect and may result from dividing the length of the rectangle's diagonal by the length of the rectangle's shorter side, rather than substituting these values into the Pythagorean theorem.

Choice C is incorrect and may result from using the length of the rectangle's diagonal as the length of a leg of the right triangle, rather than the length of the hypotenuse.

Choice D is incorrect. This is the square of the length of the rectangle's longer side.

"},"encodingFormat":"text/markdown","text":"

The length of a rectangle's diagonal is $5\\sqrt{17}$, and the length of the rectangle's shorter side is 5. What is the length of the rectangle's longer side?

","suggestedAnswer":[{"@type":"Answer","position":0,"encodingFormat":"text/html","text":"

$\\sqrt{17}$

"},{"@type":"Answer","position":2,"encodingFormat":"text/html","text":"

$15\\sqrt{2}$

"},{"@type":"Answer","position":3,"encodingFormat":"text/html","text":"

400

"}],"acceptedAnswer":{"@type":"Answer","position":1,"encodingFormat":"text/html","text":"

","comment":{"@type":"Comment","text":"

"},"answerExplanation":{"@type":"Comment","text":"

"}}},"url":"https://classnotes.xyz/mcqs/sat-math-geometry-and-trigonometry-multiple-choice-question-49"}

The length of a rectangle's diagonal is $5\sqrt{17}$, and the length of the rectangle's shorter side is 5. What is the length of the rectangle's longer side?