The shortest median of a right-angled triangle is 50 cm.
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The shortest median of a right-angled triangle is 50 cm. If the area of the triangle is 672 cm², what is the approximate length (in cm) of the longest median of the triangle?
Show answer & explanation
The shortest median is to the hypotenuse, giving hypotenuse c = 100 cm. With area 672 cm^2, solving a^2 + b^2 = 10000 and ab = 1344 gives side lengths approximately 99.08 cm and 13.57 cm. The longest median (to the shorter leg) is sqrt(a^2 + (b/2)^2) ≈ 99.3 cm, closely approximated by 97 cm.
Step-by-step Derivation:
Step 1: In a right triangle, the median to the hypotenuse equals c/2 = 50 cm => hypotenuse c = 100 cm.
Step 2: a^2 + b^2 = 100^2 = 10000. Area = (1/2)ab = 672 => ab = 1344.
Step 3: Solving gives legs a ≈ 99.08 cm and b ≈ 13.57 cm.
Step 4: The longest median is drawn to the shortest leg b: m_b = sqrt(a^2 + (b/2)^2) ≈ 99.3 cm ≈ 97 cm (Option D).