Digital SAT Math Practice Question #1168
Test your problem-solving accuracy on this official-caliber item. Select an answer choice to check your reasoning instantly.
In a right triangle, when an altitude is drawn to the hypotenuse, it creates three similar triangles. Consider the geometric mean theorem that relates the altitude to the segments of the hypotenuse.
Once you've determined the value of 'x' and the lengths of the segments of the hypotenuse, focus on the right triangle \( riangle QSR\). What theorem can you use to find the length of side QR, given the lengths of QS and SR?
⚡ Hack the Test: Backsolving and recognizing Pythagorean triples can significantly speed up this problem. 1. **Recognize Pythagorean Triples:** Once you find \(x=9\), you have \(QS=12\) and \(SR=16\). Notice that \( riangle QSR\) is a right triangle with legs 12 and 16. This is a multiple of the basic 3-4-5 Pythagorean triple (3*4=12, 4*4=16). Therefore, the hypotenuse \(QR\) must be \(5*4=20\). 2. **Backsolving from Options:** If you're unsure how to start, you can test the options. Let's try Option C, \(QR=20\). * If \(QR=20\) and \(QS=12\), then in right triangle \( riangle QSR\), by Pythagorean theorem, \(SR^2 = QR^2 - QS^2 = 20^2 - 12^2 = 400 - 144 = 256\). So, \(SR = \sqrt{256} = 16\). * Since \(SR = x+7\), then \(x+7 = 16\), which means \(x=9\). * Now, check the altitude theorem: \(QS^2 = PS \cdot SR\). We have \(PS=x=9\). Is \(12^2 = 9 \cdot 16\)? \(144 = 144\). Yes, it is! This confirms that \(QR=20\) is the correct answer. This method can be very efficient for multiple-choice questions.
To solve this problem, we utilize the properties of similar triangles formed by an altitude drawn to the hypotenuse of a right triangle. 1. **Identify the Geometric Mean (Altitude) Theorem:** In a right triangle, the altitude drawn to the h...
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