Question
A right triangle is drawn on a coordinate plane with a line drawn from the right angle that is perpendicular to the hypotenuse. The hypotenuse consists of a short line segment measuring 2 units and a long line segment measuring 30 units. Using the geometric mean, what is the distance of the short leg of the original triangle?(1 point) Responses 5.3 units 5.3 units 28 units 28 units 8 units 8 units 60 units
Answers
GPT-4o mini
In the context of a right triangle with a line drawn from the right angle that is perpendicular to the hypotenuse, we can use the geometric mean to find the lengths of the legs of the triangle.
Let's denote the lengths of the two segments of the hypotenuse as \( a = 2 \) units and \( b = 30 \) units. The formula for the length of the leg perpendicular to the hypotenuse (short leg) in terms of the segments \( a \) and \( b \) is given by:
\[
\text{Leg} = \sqrt{a \times b}
\]
Substituting the values of \( a \) and \( b \) into the formula:
\[
\text{Leg} = \sqrt{2 \times 30} = \sqrt{60}
\]
Calculating \( \sqrt{60} \):
\[
\sqrt{60} \approx 7.75
\]
Among the multiple choices provided, the one closest to \( 7.75 \) is **8 units**.
Thus, the distance of the short leg of the original triangle is **8 units**.
Let's denote the lengths of the two segments of the hypotenuse as \( a = 2 \) units and \( b = 30 \) units. The formula for the length of the leg perpendicular to the hypotenuse (short leg) in terms of the segments \( a \) and \( b \) is given by:
\[
\text{Leg} = \sqrt{a \times b}
\]
Substituting the values of \( a \) and \( b \) into the formula:
\[
\text{Leg} = \sqrt{2 \times 30} = \sqrt{60}
\]
Calculating \( \sqrt{60} \):
\[
\sqrt{60} \approx 7.75
\]
Among the multiple choices provided, the one closest to \( 7.75 \) is **8 units**.
Thus, the distance of the short leg of the original triangle is **8 units**.