Question

State whether each sentence is true or false. If false, replace the underlined word, phrase, expression, or number to make a true sentence.

A triangle with sides having measures of 3, 4 and 6is a right triangle.

Short answer

The given sentence is False.

The true sentence is A triangle with sides having measures of 3, 4, and 5 is a right triangle.

Step-by-step solution

Step 1. Determine whether the given sentence is true or false.

If c is the longest side of the triangle, a and b are the other two sides of the triangle then:

(i) If $c^2<a^2+b^2$, then the triangle is an acute triangle.

(ii) If $c^2=a^2+b^2$, then the triangle is a right triangle.

(iii) If $c^2>a^2+b^2$, then the triangle is an obtuse triangle.

Step 2. Checking the category of the given triangle.

The triangle is having sides of measures 3, 4, and 6.

The longest side of the triangle is 6.

Therefore, the value of c is 6.

Therefore, the values of a and b are 3 and 4 respectively.

Now, it can be obtained that:

$\begin{array}{c}{a}^2=3^2=9\\ b^2=4^2=16\\ c^2=6^2=36\\ a^2+b^2=9+16=25\end{array}$

It can be noticed that:

$$\begin{gathered} 25<36 \\ {a}^2+b^2<36 \\ {a}^2+b^2<c^2 \end{gathered}$$

As, $c^2>a^2+b^2$, therefore, the triangle is an obtuse triangle.

Therefore, the given sentence is false.

Step 3. Write the true sentence.

If $c=5$,$a=3$ and $b=4$;

$\begin{array}{c}{a}^2=3^2=9\\ b^2=4^2=16\\ c^2=5^2=25\\ a^2+b^2=9+16=25=c^2\end{array}$

As, $c^2=a^2+b^2$, when $c=5$,$a=3$ and $b=4$, the triangle is a right triangle.

Therefore, the true sentence is a triangle with sides having measures of 3, 4, and 5 is a right triangle.