Question

Can someone help me with this geometry question I don’t know if I am right.

Answer 1

Let us find out if the given two triangles ABC and DEF are similar triangles or not.

Triangle ABC is a right-angled triangle so we can apply the Pythagorean theorem to find the missing side.

`a^2+b^2=c^2`

Where a and b are the shorter sides and c is the longest side (hypotenuse)

`\begin{gathered} 20^2+21^2=c^2 \\ 400+441=c^2 \\ 841=c^2 \\ \sqrt[]{841}=c \\ 29=c \\ c=29 \end{gathered}`

Similarly, we can apply the Pythagorean theorem to triangle DEF to find the missing side.

`\begin{gathered} d^2+e^2=f^2 \\ 40^2+e^2=58^2 \\ e^2=58^2-40^2 \\ e^2=3364-1600 \\ e^2=1764 \\ e=\sqrt[]{1764} \\ e=42 \end{gathered}`

Now, recall that two triangles are similar if the ratio of the corresponding sides is equal.

The corresponding sides are

AB = DE

BC = EF

AC = DF

`\begin{gathered} (DE)/(AB)=(EF)/(BC)=(DF)/(AC) \\ (40)/(20)=(42)/(21)=(58)/(29) \\ (2)/(1)=(2)/(1)=(2)/(1) \end{gathered}`

As you can see, the ratio of the corresponding sides of the two triangles is equal.

Hence, the triangles ABC and DEF are similar.