One side of a triangle measured 126 m and the difference in lengths of its hypotenuse and other side is - Maths - Heron\s Formula

Question

One side of a triangle measured 126 m and the difference in
lengths of its hypotenuse and other side is 42 m Find the measure
of its two unknown sides and calculate its area Verify the result
using herons formula

Manbar Singh · Accepted Answer

In right angled △ ABC, ∠B = 90°, AC is a hypotenuse, AB and BC are other two sides
then, BC = 126 mand AC - AB = 42 m  
1In right angled △ ABCAC2  = AB2 + BC2    By pythagoras theorem⇒ AC2 - AB2 = BC2⇒ AC + AB AC - AB = 1262⇒ 42 AC + AB = 126 × 126     As AC - AB = 42 m⇒ AC + AB = 126 × 12642  = 3 × 126⇒ AC + AB = 378  
2Adding 1 and 2, we get,2 AC = 420⇒ AC = 210 mPutting the value of AC in 1, we get,210 - AB = 42⇒ 210 - 42 = AB⇒ AB = 168 mTherefore, area of △ ABC = 12 × base × height                                         = 12 × BC × AB                                         = 12 × 126 ×168                                         = 63 × 168 m2                                         = 10584 m2

Area of △ ABC by Heron's formula:Let BC = a, AC = b  and   AB = cLet s be the semiprimeter of △ ABC, then, s = a + b + c2 = 126 + 210 + 1682 = 5042 = 252 mNow area of △ ABC = s s - a s - b s - c                                = 252 252 - 126 252 - 210 252 - 168                                = 252 × 126 × 42 × 84                                 = 7 × 36 × 7 × 18 × 7 × 6 × 7 × 12                                = 7 × 3 × 12 × 7 × 6 ×3 × 7 × 6 × 7 × 12                                = 7 × 7 × 12 × 3 × 6                                = 10584 m2

One side of a triangle measured 126 m and the difference in lengths of its hypotenuse and other side is 42 m. Find the measure of its two unknown sides and calculate its area. Verify the result using herons formula