Question

(a) 50 cm, 40 cm, 80 cm
triangle possible
triangles with these measurements
no triangle possible
(b) 9 m, 4 m, 4 m
triangle possible
triangles with these measurements
no triangle possible
(c) 54°, 36°, 52°
triangle possible
triangles with these measurements
no triangle possible

Explanation:

Part (a)

Step 1: Check Triangle Inequality

For a triangle with side lengths \(a\), \(b\), \(c\), the sum of any two sides must be greater than the third side. Here, \(a = 50\space\text{cm}\), \(b = 40\space\text{cm}\), \(c = 80\space\text{cm}\).
Check \(a + b\): \(50 + 40 = 90\space\text{cm}\). Since \(90>80\), \(a + c\): \(50 + 80 = 130>40\), \(b + c\): \(40 + 80 = 120>50\). Wait, but wait, actually, \(50 + 40 = 90>80\), but wait, no—wait, the triangle inequality is that the sum of the two smaller sides must be greater than the largest side. The two smaller sides are \(40\) and \(50\), sum is \(90\), which is greater than \(80\). Wait, but wait, maybe I made a mistake. Wait, \(50 + 40 = 90\), which is greater than \(80\), \(50 + 80 = 130>40\), \(40 + 80 = 120>50\). Wait, but the answer here—wait, no, maybe I miscalculated. Wait, \(40 + 50 = 90\), which is greater than \(80\), so triangle should be possible? Wait, no, wait the problem—wait, maybe the user made a typo? Wait, no, let's recheck. \(40 + 50 = 90\), \(90>80\), so triangle is possible? Wait, but the options are "Triangle possible" or "No triangle possible". Wait, maybe I messed up. Wait, no, the triangle inequality: the sum of any two sides must be greater than the third. So \(40 + 50 = 90>80\), \(40 + 80 = 120>50\), \(50 + 80 = 130>40\). So triangle is possible? Wait, but the original problem—wait, maybe I misread. Wait, \(50\), \(40\), \(80\): \(40 + 50 = 90>80\), so yes, triangle is possible? Wait, no, wait, maybe the user intended different numbers? Wait, no, the given is \(50\), \(40\), \(80\). So according to triangle inequality, it is possible. Wait, but maybe I made a mistake. Wait, no, \(40 + 50 = 90>80\), so triangle is possible. Wait, but the options—wait, maybe the problem is different. Wait, no, let's check again. \(50 + 40 = 90\), which is greater than \(80\), so triangle is possible.

Wait, no, wait, maybe I made a mistake. Wait, \(40 + 50 = 90\), which is greater than \(80\), so the triangle inequality holds. So "Triangle possible".

Part (b)

Step 1: Check Triangle Inequality (Isosceles Triangle)

Sides: \(9\space\text{m}\), \(4\space\text{m}\), \(4\space\text{m}\). The two smaller sides are \(4\) and \(4\), sum is \(4 + 4 = 8\space\text{m}\). The largest side is \(9\space\text{m}\). Since \(8<9\), the sum of the two smaller sides is not greater than the largest side. So triangle inequality fails. Thus, "No triangle possible".

Part (c)

Step 1: Check Angle Sum Property

The sum of the interior angles of a triangle must be \(180^\circ\). Given angles: \(54^\circ\), \(36^\circ\), \(52^\circ\). Sum: \(54 + 36 + 52 = 142^\circ\). Since \(142^\circ
eq180^\circ\), the angle sum property is not satisfied. Thus, "No triangle possible".

Answer:

(a) Triangle possible
(b) No triangle possible
(c) No triangle possible