Question

Which dimensions can create only one unique triangle?

Answer 1

Answer:

Three sides measuring 5 in., 12 in., and 14 in.

Explanation:

First of all, you need to know some facts about every triangle:

FIRST FACT:

If two sides of a triangle are $a \ and \ b$, then the third side must be less than $a+b$. Why? because if its length is exactly $a+b$, then you can't build up a triangle, but you'll have a line segment. In other words:

$a$

SECOND FACT:

The internal angles of any triangle add up to 180°. So if ∠A, ∠B and ∠C are the internal angles of a triangle, then:

$\angle A +\angle B+ \angle C=180^{\circ}$

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So let's analyze each case:

Case 1. Three angles measuring 25 degrees, 65 degrees, and 90 degrees :

$\angle A=25^{\circ} \\ \\ \angle B=65^{\circ} \\ \\ \angle C=90^{\circ} \\ \\ \\ \angle A +\angle B+ \angle C=25^{\circ}+65^{\circ}+90^{\circ}=185^{\circ} \neq 180^{\circ}$

The sum is greater than 180 degrees, so these angles can't form a triangle

Case 2. Three angles measuring 50 degrees, 50 degrees, and 50 degrees

$\angle A=50^{\circ} \\ \\ \angle B=50^{\circ} \\ \\ \angle C=50^{\circ} \\ \\ \\ \angle A +\angle B+ \angle C=50^{\circ}+50^{\circ}+50^{\circ}=150^{\circ} \neq 180^{\circ}$

The sum is less than 180 degrees, so these angles can't form a triangle

Case 3. Three sides measuring 5 in., 12 in., and 14 in.

$a=5 \\ \\ b=12 \\ \\ c=14 \\ \\ \\ 5$

The inequalities are true, so these sides form a triangle.

Case 4: Three sides measuring 4 ft, 8 ft, and 14 ft

$a=4 \\ \\ b=8 \\ \\ c=14 \\ \\ \\ 4$

Since the third inequality is not true, then we can't form a triangle with these sides.