Question

The lengths of the sides of a triangle are and

1. Find the value (s) of that make the triangle isosceles.
2. Does any value of make the triangle equilateral? Explain.

Short answer

1. The value of $t$that makes the triangle isosceles is 6.
2. There is no value of$t$ that will make the triangle equilateral.

Step-by-step solution

Part a. Step 1. Description of step.

In isosceles triangle, two sides of the triangle are equal.

$$\begin{gathered} 3t=5t-12 \\ 2t=12 \end{gathered}$$

Part a. Step 2. Description of a step.

Divide each side of the obtained equation in step 1 by 2 to find the value of $t$.

$\begin{array}{c}\frac{2t}{2}=\frac{12}{2}\\ t=6\end{array}$

Part a. Step 3. Description of a step.

Substitute 6 for $t$into $3t$to find the length of first side.

$$\begin{gathered} 3t=3\times 6 \\ =18 \end{gathered}$$

Part a. Step 4. Description of a step.

Substitute 6 for$t$ into$5t-12$ to find the length of second side.

$$\begin{gathered} 5t-12=5\times 6-12 \\ =30-12 \\ =18 \end{gathered}$$

Part a. Step 5. Description of a step.

Substitute 6 for$t$into$t+20$to find the length of third side.

width="98" height="44" role="math">t+20=6+20=26

The three sides of the triangle are 18, 18 and 26. Therefore, the value of$t$ that makes the triangle isosceles is 6.

Part b. Step 1. Apply the definition of an equilateral triangle.

An equilateral triangle is a triangle in which all the three sides have the same length.

$$\begin{gathered} 3t=5t-12 \\ 2t=12 \end{gathered}$$

Part b. Step 2. Description of a step.

Divide each side of the obtained equation in step 1 by 2 to find the value of $t$.

$\begin{array}{c}\frac{2t}{2}=\frac{12}{2}\\ t=6\end{array}$

Part b. Step 3. Description of a step.

Substitute 6 for $t$into $3t$to find the length of first side.

$\begin{array}{c}3t=3\times 6\\ =18\end{array}$

Part b. Step 4. Description of a step.

Substitute 6 for $t$into$5t-12$ to find the length of second side.

$$\begin{gathered} 5t-12=5\times 6-12 \\ =30-12 \\ =18 \end{gathered}$$

Part b. Step 5. Description of a step.

Substitute 6 for$t$into$t+20$to find the length of third side.

$$\begin{gathered} t+20=6+20 \\ =26 \end{gathered}$$

The three sides of the triangle are 18, 18 and 26, which does not represent an equilateral triangle. Therefore, there is no value of$t$that will make this triangle equilateral.