The area of a right triangle is 84 square cm, and one perpendicular side is 5 cm longer than the other. What is the length of the shorter perpendicular side?
Answer and explanation
Correct answer: \(\frac{\sqrt{697}-5}{2}\) cm
Let the shorter perpendicular side be \(x\) cm. Then the other side is \(x+5\) cm. Using the area formula, \(\frac{1}{2}x(x+5)=84\), which gives \(x^2+5x-168=0\). Applying the quadratic formula gives \(x=\frac{-5\pm\sqrt{697}}{2}\). Since a length must be positive, the valid answer is \(x=\frac{\sqrt{697}-5}{2}\) cm, approximately 10.69 cm. The 12 cm option is incorrect because the corresponding sides would be 12 cm and 17 cm, giving an area of 102 square cm. In an exam, reject the negative root as a physical length.
Frequently asked questions
What is the correct answer to this question?
\(\frac{\sqrt{697}-5}{2}\) cm
Why is this the correct answer?
Let the shorter perpendicular side be \(x\) cm. Then the other side is \(x+5\) cm. Using the area formula, \(\frac{1}{2}x(x+5)=84\), which gives \(x^2+5x-168=0\). Applying the quadratic formula gives \(x=\frac{-5\pm\sqrt{697}}{2}\). Since a length must be positive, the valid answer is \(x=\frac{\sqrt{697}-5}{2}\) cm, approximately 10.69 cm. The 12 cm option is incorrect because the corresponding sides would be 12 cm and 17 cm, giving an area of 102 square cm. In an exam, reject the negative root as a physical length.
Which subject and chapter does this question cover?
This is a Class 10 Mathematics question. Chapter: Quadratic Equations. Topic: Word Problems and Applications.
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