Chapter 12.8, Problem 24E

To determine

To find: The theorem that would be easier to prove with a coordinate proof than other type and also write a coordinate proof.

Answer to Problem 24E

The theorem easier to prove with a coordinate axis is The Base Angle Theorem.

Explanation of Solution

Given information:

  

The vertex of triangle $ABC$ are,

  $\begin{array}{l}A\left(-h,0\right)\\ B\left(0,k\right)\\ C\left(h,0\right)\end{array}$

In a triangle $ABC$ the sides $AB$ and $AC$ are congruent.

Now the slope of is given by the formula,

  $m=\frac{y_2-y_1}{x_2-x_1}$

Thus, the slope of $BC$ is given by substituting the value of $x_{1,}{x}_2,y_1,y_2$ in the given formula.

Here,

  $x_1=0,x_2=h,y_1=k,y_2=0$

Thus,

  $\begin{array}{c}{m}_1=\frac{0-k}{h-0}\\ =\frac{-k}{h}\end{array}$

Similarly the slope of $AB$ is

  $\begin{array}{c}{m}_2=\frac{k-0}{0-\left(-h\right)}\\ =\frac{k}{h}\end{array}$

Hence,

Slope of $BC$ is the negative of slope of $AB$ .

  $m_1=-m_2$

The theorem that would be easier to prove with a coordinate proof is the Base Angle Theorem. It states that If two sides of a triangle are congruent, then the opposite angles of these sides are also congruent. So,

  $<BCA\cong <BAC$

Therefore, the theorem that would be easier to prove is The Base Angle Theorem.