A right triangle has sides 6, a and 10. If the area of the triangle i

I. Right triangle ratio rule

Another 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 theorem
Alternatively, 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

Recall that the sum of the lengths of the two shorter sides of a triangle must be greater than the length of the longest side; thus, a could be 5 (since 6 + 5 > 10) or 8 (since 6 + 8 > 10) or 11.6 (since 6 + 10 > 11.6). We can eliminate choices A and B.

However, a could not be 5 since the triangle is a right triangle and 6^2 + 5^2 ≠ 10^2. We can eliminate answer choice C. On the other hand, a could be 8 since 6^2 + 8^2 = 10^2 or a could be 11.6 since 6^2 + 100^2 = 11.6^2 (if we ignore round-off error).

Lastly, the area of a right triangle is ½ the product of the lengths of its two legs. If a = 8, then the area of the triangle is ½(6)(8) = 24. Similarly, if a = 10, then the area of the triangle is ½(6)(10) = 30. However, since we are given that the area of the triangle is less than 30, a must be 8.

Answer: D