Question

^LMN with L(-6, -2), M(2, 4), and N(8, -4), and ^PQR with P(3, 1), Q(-1, -2), and R(-2, 2) The triangles are similar. The ratio of their corresponding sides is ____

Answer 1

Solution

For the Big triangle

We will find the size of each sides of the triangle

The formula for finding the distance between two points (x1, y1) and (x2, y2) is given by

$d=\sqrt[]{(x_2-x_1)^2+(y_2-y_1)^2}$

To find n we have the points (-6, -2) and (2, 4)

$\begin{gathered} n=\sqrt[]{(2-(-6))^2+(4-(-2))^2_{}} \\ n=\sqrt[]{(2+6)^2+(4+2)^2} \\ n=\sqrt[]{8^2+6^2} \\ n=\sqrt[]{64+36} \\ n=\sqrt[]{100} \\ n=10 \end{gathered}$

We find l we have the points (8, -4) and (2, 4)

$\begin{gathered} l=\sqrt[]{(2-8)^2+(4-(-4))^2} \\ l=\sqrt[]{6^2+8^2} \\ l=\sqrt[]{100} \\ l=10 \end{gathered}$

To find m we have the points (-6, -2) and (8, -4)

$\begin{gathered} m=\sqrt[]{(8-(-6))^2+(-4-(-2))^2} \\ m=\sqrt[]{(8+6)^2+(-4+2)^2} \\ m=\sqrt[]{14^2+2^2} \\ m=\sqrt[]{196+4} \\ m=\sqrt[]{200} \\ m=10\sqrt[]{2} \end{gathered}$

For the Smaller triangle

We will find all sides as well

For p we have the points (-2, 2) and (-1, -2)

$\begin{gathered} p=\sqrt[]{(-1-(-2))^2+(-2-2)^2} \\ p=\sqrt[]{(-1+2)^2+(-4)^2} \\ p=\sqrt[]{1^2+4^2} \\ p=\sqrt[]{1+16} \\ p=\sqrt[]{17} \end{gathered}$

For r we have the points (-1, -2) and (3, 1)

$\begin{gathered} r=\sqrt[]{(3-(-1))^2+(1-(-2))^2} \\ r=\sqrt[]{(3+1)^2+(1+2)^2} \\ r=\sqrt[]{4^2+3^2} \\ r=\sqrt[]{16+9} \\ r=\sqrt[]{25} \\ r=5 \end{gathered}$

For q we have the points (-2, 2) and (3, 1)

$\begin{gathered} q=\sqrt[]{(3-(-2))^2+(1-2)^2} \\ q=\sqrt[]{(3+2)^2+(-1)^2} \\ q=\sqrt[]{5^2+1} \\ q=\sqrt[]{25+1} \\ q=\sqrt[]{26} \end{gathered}$

Comparing the two triangles side by sides

Comparing each ratios

They are NOT similar beacause the ratio of each sides are not same

$\begin{gathered} RatioOfFirstSide=\frac{10}{5}=2 \\ RatioOfSecondSide=\frac{10}{\sqrt[]{17}}=2.425 \\ RatioOfThirdSide=\frac{10\sqrt[]{2}}{\sqrt[]{26}}=2.774 \\ \\ \frac{10}{5}\ne\frac{10}{\sqrt[]{17}}\ne\frac{10\sqrt[]{2}}{\sqrt[]{26}} \end{gathered}$

THEY ARE NOT SIMILAR TRIANGLE