    Question

# $△ABC$ with vertices$A\left(-2,0\right),B\left(2,0\right)$ and$C\left(0,2\right)$ is similar to $△DEF$ with vertices $D\left(-4,0\right)E\left(4,0\right)$ and $F\left(0,4\right)$. State whether the following statement is true or false. Justify your answer.

A
True
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B
False
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Solution

## The correct option is A True Step 1: Find the length of sides of $△\mathrm{ABC}$We know that Distance formula,$d=\sqrt{{\left({x}_{2}–{x}_{1}\right)}^{2}+{\left({y}_{2}–{y}_{1}\right)}^{2}}$In triangle $△\mathrm{ABC}$by applying the distance formula$\begin{array}{rcl}AB& =& \sqrt{\left(\left(2-{\left(-2\right)\right)}^{2}+0}\\ & =& \sqrt{{\left(2+2\right)}^{2}}\\ & =& \sqrt{{4}^{2}}\\ & =& \sqrt{16}\\ & =& 4\end{array}$$\begin{array}{rcl}\mathrm{BC}& =& \sqrt{{\left(0-2\right)}^{2}+{\left(2-0\right)}^{2}}\\ & =& \sqrt{4+4}\\ & =& 2\sqrt{2}\end{array}$$CA=\sqrt{{\left(-2-0\right)}^{2}+{\left(0-2\right)}^{2}}\phantom{\rule{0ex}{0ex}}=\sqrt{4+4}\phantom{\rule{0ex}{0ex}}=2\sqrt{2}$Step 2: Find the length of sides of $△DEF$Now in $△DEF$$\begin{array}{rcl}\mathrm{DE}& =& \sqrt{\left(4-{\left(-4\right)\right)}^{2}+0}\\ & =& \sqrt{{\left(4+4\right)}^{2}}\\ & =& \sqrt{64}\\ & =& 8\end{array}$$\begin{array}{rcl}EF& =& \sqrt{{\left(0-4\right)}^{2}+{\left(4-0\right)}^{2}}\\ & =& \sqrt{16+16}\\ & =& 4\sqrt{2}\end{array}$$\begin{array}{rcl}FD& =& \sqrt{{\left(-4-0\right)}^{2}+{\left(0-4\right)}^{2}}\\ & =& \sqrt{16+16}\\ & =& 4\sqrt{2}\end{array}$Step 2: Find the ratio of corresponding sideswe have the ratio of corresponding sides of the triangle ,$\frac{AB}{DE}=\frac{BC}{EF}=\frac{CA}{FD}=\frac{1}{2}$$\therefore \Delta ABC~\Delta DEF$Hence, triangle $△ABC$ and$△DEF$ are similar.  Suggest Corrections  0      Similar questions  Explore more