Can you construct a triangle with sidelengths in the ratio 1 : 3 : 5? Why or why not?
No, a triangle with sidelengths in the ratio 1:3:5 cannot be constructed.
Reason: Triangle Inequality Property states that the sum of any two sides must be greater than the third side.
Check: Sum of two smaller sides = 1 + 3 = 4 4 < 5 (largest side)
∴ Triangle cannot be formed because the sum of the two smaller sides is less than the largest side.
Explanation
The chapter introduces triangle construction with given ratios. The triangle inequality theorem states that for any valid triangle, the sum of any two sides must exceed the third side. The ratio 1:3:5 violates this condition because 1+3=4<5, making construction impossible.
Solution Steps
State the Triangle Inequality Property: the sum of any two sides must be greater than the third side.
Add the two smaller sides: 1 + 3 = 4.
Compare the sum to the largest side: 4 < 5.
Conclude that the triangle cannot be formed since the sum of two smaller sides is less than the largest side.