Question

consider triangle ghj.
what is the length of line segment hj?
(options: 5 units, $10\sqrt{3}$ units, 10 units, $5\sqrt{3}$ units)

Explanation:

Step1: Identify the triangle type

Triangle \( GHJ \) is a right - triangle with \( \angle H = 90^{\circ} \), \( GJ = 10 \) units (hypotenuse) and \( GH = 5 \) units (one leg). Let \( HJ=x \) (the other leg).

Step2: Apply the Pythagorean theorem

The Pythagorean theorem states that for a right - triangle with legs \( a \), \( b \) and hypotenuse \( c \), \( a^{2}+b^{2}=c^{2} \). In triangle \( GHJ \), \( GH^{2}+HJ^{2}=GJ^{2} \). Substituting the known values: \( 5^{2}+x^{2}=10^{2} \).

Step3: Solve for \( x \)

First, expand the equation: \( 25 + x^{2}=100 \). Then, subtract 25 from both sides: \( x^{2}=100 - 25=75 \). Take the square root of both sides: \( x=\sqrt{75}=\sqrt{25\times3}=5\sqrt{3} \).

Answer:

\( 5\sqrt{3} \) units