Gradient and Similar Triangles
We've already looked at similar shapes and focussed on similar triangles. Most importantly, we learnt that in similar shapes, all the corresponding sides are in the same ratio and all corresponding angles are equal.
However, similar shapes are not aways drawn as two distinct shapes. They can also be graphed on number planes.
We still look for corresponding sides to be in the same ratio, as well as equal corresponding angles to determine whether shapes on a number plane are similar.
What's really cool is that triangles made by drawing a vertical line and a horiztonal line from a line we've graphed will always be similar and the ratio of the sides will be the gradient.
We find the gradient by using the formula:
$m=\frac{rise}{run}$m=riserun
or
$m=\frac{y_2-y_1}{x_2-x_1}$m=y2−y1x2−x1
Let's tie in some previous understandings we developed about triangles, gradients and angles.
This mathlet shows you a line (you can change the gradient of it) and a number of triangles that are formed from it (you can adjust them). Take particular note of the angles in the triangles. Are the triangles similar?
- The value of the gradient can be found using $\frac{rise}{run}$riserun
- The angle, (sometimes called the angle of inclination), can be found using $\tan\theta=\frac{\text{opposite }}{\text{adjacent }}$tanθ=opposite adjacent
- In this diagram, $\frac{rise}{run}$riserun is the same calculation as $\frac{\text{opposite }}{\text{adjacent }}$opposite adjacent
- Any right triangle with the same gradient, will also have the same angle of inclination
From our mathlet we can see that all the triangles have a right angle, and the same angle of inclination they also have the same value for the third angle. See how all the triangles have the same sized angles, this is one of our definitions of similarity. So all triangles formed (in this manner) from a line with the same gradient are similar.
Worked Examples
Question 1
Consider the points $A$A, $B$B and $C$C.
Complete the directions that explain how to move from point $A$A to point $B$B.
From $A$A, move $\editable{}$ units up and $\editable{}$ units to the right.
Express the direction of the movement in the previous question as a simplified ratio, comparing vertical movement to horizontal movement. Express the ratio in the form $a:b$a:b.
$\editable{}:\editable{}$:
Complete the directions that explain how to move from point $A$A to point $C$C.
From $A$A, move $\editable{}$ units up and $\editable{}$ units to the right.
Express the direction of the movement in the previous question as a simplified ratio, comparing vertical movement to horizontal movement. Express the ratio in the form $a:b$a:b.
$\editable{}:\editable{}$:
Question 2
Which of the triangles listed below are similar to $\triangle FDB$△FDB?
$\triangle EFA$△EFA
A$\triangle CBA$△CBA
B$\triangle AEB$△AEB
C$\triangle CFB$△CFB
D