Bunuel wrote:
A right triangle has sides 6, a and 10. If the area of the triangle is less than 30, what is the value of a?
A. 3
B. 4
C. 5
D. 8
E. 11.6
I. Right triangle ratio ruleAnother way: use a property of 3-4-5 right triangles, listed below.
Because the area of the right triangle is less than 30, the side with length 10 is the hypotenuse.
If the given sides were legs (which, in a right triangle, are base and height), area would be \(\frac{(6 * 10)}{2}=30\). Not allowed.
\(a\) cannot be greater than 10. If \(a\) were greater than 10, \(a\) would be the hypotenuse, and area would \(=30\)
The side with length of 10 must be the hypotenuse. The side with length of 6 is a leg.
Rule: in a right triangle, if
one leg and the
hypotenuse are in the ratio \(3x:5x\) or \(4x:5x\), the triangle is a \(3x-4x-5x\) right triangle.
In this instance, the ratio of leg to hypotenuse is \(3x:5x\)
\(3x=6\), so \(x=2\) (or \(5x=10\), so \(x=2\))
\(a\) = the missing ratio part from the rule: \(4x\)
Leg \(a=(4*2)=8\)
Answer D
Pythagorean theoremAlternatively, use area limit and the Pythagorean theorem to find \(a\)
Area, A = \(\frac{b*h}{2}<30\). If the given sides were legs, area would = \(\frac{6*10}{2}=30\)
\(a\) cannot be greater than 10. If \(a\) were greater than 10, \(a\) would be the hypotenuse, and area would \(=30\)
The side with length of 6 is a leg
The side with length of 10 is the hypotenuse
Pythagorean theorem: \((6^2+ a^2)=10^2\)
\(a^2=(100-36)\)
\(a^2=64\)
\(a=8\)
Answer D
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