Page 5 - ln1

Show that in a right angled triangle, the hypotenuse is the longest side.

Solution: Let us consider a right-angled triangle ABC, right-angled at B.
In ΔABC,
∠A + ∠B + ∠C = 180° (Angle sum property of a triangle)
∠A + 90º + ∠C = 180°
∠A + ∠C = 90°
Hence, the other two angles have to be acute (i.e., less than 90º).
∠B is the largest angle in ΔABC.
∠B >∠A and ∠B >∠C
AC > BC and AC > AB [In any triangle, the side opposite to the larger (greater) angle is
longer.]
Therefore, AC is the largest side in ΔABC.
However, AC is the hypotenuse of ΔABC. Therefore, hypotenuse is the longest side in a
right-angled triangle.

Question 2 In the given figure sides AB and AC of ΔABC are extended to points P
and Q respectively. Also, ∠PBC <∠QCB. Show that AC > AB.

Solution:In the given figure,
∠ABC + ∠PBC = 180° (Linear pair)
∠ABC = 180° − ∠PBC ... (1)
Also,∠ACB + ∠QCB = 180°
∠ACB = 180° − ∠QCB … (2)
As ∠PBC <∠QCB,
180º − ∠PBC > 180º − ∠QCB
∠ABC >∠ACB [From Equations (1) and (2)]
AC > AB (Side opposite to the larger angle is larger.)
Hence proved, AC > AB

Question 3. In the given figure, AD ⊥ BC and CD >BD. Show that AC > AB.