Question

1. a right triangle has one leg that is 7 less than 3 times the other. if the hypotenuse is 94 m, how long are the two legs?

Explanation:

Step1: Define Variables

Let one leg be \( x \) meters. Then the other leg is \( 3x - 7 \) meters.

Step2: Apply Pythagorean Theorem

For a right triangle, \( a^2 + b^2 = c^2 \), where \( c \) is the hypotenuse. So \( x^2 + (3x - 7)^2 = 94^2 \).

Step3: Expand and Simplify

\( x^2 + 9x^2 - 42x + 49 = 8836 \)
\( 10x^2 - 42x + 49 - 8836 = 0 \)
\( 10x^2 - 42x - 8787 = 0 \)
Divide by common factor (if any, here none obvious), use quadratic formula \( x=\frac{-b\pm\sqrt{b^2 - 4ac}}{2a} \), where \( a = 10 \), \( b = -42 \), \( c = -8787 \).
Discriminant \( D = (-42)^2 - 4(10)(-8787) = 1764 + 351480 = 353244 \).
\( \sqrt{D} = 594 \).
\( x=\frac{42\pm594}{20} \).
We take the positive root: \( x=\frac{42 + 594}{20}=\frac{636}{20}=31.8 \) (wait, maybe miscalculation. Wait, 3x -7 must be positive. Wait, let's check again. Wait, 94 squared is 8836. Let's re - expand \( (3x - 7)^2 = 9x^2 - 42x + 49 \). So equation is \( x^2+9x^2 - 42x + 49 = 8836 \Rightarrow 10x^2 - 42x - 8787 = 0 \). Wait, maybe I made a mistake in discriminant. Wait, \( b^2 - 4ac=(-42)^2-4\times10\times(-8787)=1764 + 351480 = 353244 \). Square root of 353244: let's see, 594^2 = (600 - 6)^2 = 360000 - 7200 + 36 = 353236. Oh, I miscalculated. 594^2 = 353236, so discriminant is 353244, so square root is 594.006... Wait, maybe the equation is wrong. Wait, maybe the legs are integers? Let's assume one leg is \( x \), other is \( 3x - 7 \). Let's try x = 31: 3x -7 = 86. Then \( 31^2+86^2 = 961 + 7396 = 8357 eq8836 \). x = 32: 3x -7 = 89. \( 32^2 + 89^2 = 1024+7921 = 8945 eq8836 \). x = 30: 3x -7 = 83. \( 30^2+83^2 = 900 + 6889 = 7789 eq8836 \). Wait, maybe my initial variable definition is wrong? Wait, the problem says "one leg that is 7 less than 3 times the other". So maybe I mixed up. Let me re - do the quadratic formula. \( x=\frac{42\pm\sqrt{353244}}{20}=\frac{42\pm594.36}{20} \). Taking positive root: \( \frac{42 + 594.36}{20}=\frac{636.36}{20}=31.818 \). Then the other leg is \( 3\times31.818 - 7\approx95.454 - 7 = 88.454 \). Wait, but maybe I made a mistake in setting up the equation. Wait, hypotenuse is 94, so \( x^2+(3x - 7)^2 = 94^2 \). Let's compute 94^2: 90^2 + 8^2+2*90*8 = 8100 + 64 + 1440 = 9604? Wait, no! 94*94: (90 + 4)(90 + 4)=8100+720 + 16 = 8836. Yes, that's correct. Wait, 94^2 is 8836. Then where is the mistake? Wait, 3x -7, maybe the other way: one leg is 3x -7, the other is x. Let's check x = 31: 3x -7 = 86. 31²+86²=961 + 7396=8357. x=32: 3x -7=89. 32²+89²=1024 + 7921=8945. x=30: 3x -7=83. 30²+83²=900+6889=7789. x=31.8: 3x -7=88.4. 31.8²+88.4²=1011.24+7814.56=8825.8, close to 8836. x=31.83: 3x -7=88.49. 31.83²=1013.1489, 88.49²=7830.4801, sum=8843.629, close. So maybe the legs are approximately 31.8 m and 88.5 m. But maybe I made a mistake in the problem setup. Wait, maybe the problem is "7 less than 3 times" so 3x -7, and hypotenuse 94. So the quadratic solution is \( x=\frac{42\pm\sqrt{(-42)^2 - 4\times10\times(-8787)}}{2\times10}=\frac{42\pm\sqrt{1764 + 351480}}{20}=\frac{42\pm\sqrt{353244}}{20} \). Since \( \sqrt{353244}\approx594.34 \), then \( x=\frac{42 + 594.34}{20}\approx\frac{636.34}{20}=31.817 \) meters. Then the other leg is \( 3\times31.817 - 7\approx95.451 - 7 = 88.451 \) meters.

Step4: Verify

\( 31.817^2+88.451^2\approx1012.3 + 7824.6\approx8836.9\approx94^2 \). So the legs are approximately 31.8 m and 88.5 m. But maybe there is a calculation error. Wait, let's check the quadratic formula again. The quadratic is \( 10x^2 - 42x - 8787 = 0 \). Divide by GCD of 10, 42, 8787. GCD of 10 and 42 is 2, GCD of 2 and 8787: 8787 is odd, so GCD is 1. So we can't simplify. So the exact values are \( x=\frac{42 + \sqrt{353244}}{20} \) and \( 3x - 7=\frac{126 + 3\sqrt{353244}}{20}-7=\frac{126 + 3\sqrt{353244}-140}{20}=\frac{3\sqrt{353244}-14}{20} \). But maybe the problem has a typo, or I misread. Wait, the hypotenuse is 94? Maybe 95? Let's check 95^2 = 9025. Then equation \( x^2+(3x -7)^2 = 9025 \Rightarrow 10x^2 - 42x + 49 - 9025 = 0 \Rightarrow 10x^2 - 42x - 8976 = 0 \). Discriminant: \( 42^2 + 4\times10\times8976 = 1764 + 359040 = 360804 \). Square root of 360804: 600.67, no. Wait, maybe 94 is correct. So the legs are approximately 31.8 m and 88.5 m. But let's do the calculation more accurately. \( \sqrt{353244} \): 594^2 = 352836, 595^2 = 354025. So 353244 - 352836 = 408. So \( \sqrt{353244}=594+\frac{408}{2\times594}\approx594 + 0.341 = 594.341 \). Then \( x=\frac{42 + 594.341}{20}=\frac{636.341}{20}=31.817 \) meters. The other leg: \( 3\times31.817 - 7 = 95.451 - 7 = 88.451 \) meters.

Answer:

The two legs are approximately \( 31.8 \) meters and \( 88.5 \) meters. (If we consider exact form, \( \frac{42 + \sqrt{353244}}{20} \) meters and \( \frac{3\sqrt{353244}-14}{20} \) meters)