Triangular slope
Can you find the gradients of the lines that form a triangle?
Image
![Triangular slope Triangular slope](/sites/default/files/styles/large/public/thumbnails/content-id-13592-main%252520diagram.png?itok=Cc5CSq5a)
What is the gradient of OA?
What is the gradient of AB?
This problem is taken from the World Mathematics Championships
Answer: $\frac 43$ and $-\frac34$
Using scale factors
OA
Red triangles are similar so use scale factor $\frac35=0.6$ to find missing lengths
Image
![Triangular slope Triangular slope](/sites/default/files/styles/large/public/thumbnails/content-id-13592-triangular%252520slope%252520OA%2525201.png?itok=dSOQ4cVl)
Image
![Triangular slope Triangular slope](/sites/default/files/styles/large/public/thumbnails/content-id-13592-triangular%252520slope%252520OA%2525202.png?itok=sQsnzXHA)
Image
![Triangular slope Triangular slope](/sites/default/files/styles/large/public/thumbnails/content-id-13592-triangular%252520slope%252520OA%252520scale%252520factor.png?itok=tMoxUkUz)
Gradient is $\dfrac{4\times0.6}{3\times0.6}=\dfrac43$
AB
Green triangles are similar with scale factor $\frac45=0.8$
Image
![Triangular slope Triangular slope](/sites/default/files/styles/large/public/thumbnails/content-id-13592-triangular%252520slope%252520AB%2525201.png?itok=HecnDDSV)
Image
![Triangular slope Triangular slope](/sites/default/files/styles/large/public/thumbnails/content-id-13592-triangular%252520slope%252520AB%2525202.png?itok=tYJARJf3)
Image
![Triangular slope Triangular slope](/sites/default/files/styles/large/public/thumbnails/content-id-13592-triangular%252520slope%252520AB%252520scale%252520factor.png?itok=OvPHYzS-)
Gradient is $\dfrac{-3\times0.6}{4\times0.6}=-\dfrac34$
Using similar and congruent triangles
OA
Red triangles are similar so use large red triangle for the correct ratios
Image
![Triangular slope Triangular slope](/sites/default/files/styles/large/public/thumbnails/content-id-13592-triangular%252520slope%252520OA%2525201.png?itok=dSOQ4cVl)
Image
![Triangular slope Triangular slope](/sites/default/files/styles/large/public/thumbnails/content-id-13592-triangular%252520slope%252520OA%2525202.png?itok=sQsnzXHA)
Image
![Triangular slope Triangular slope](/sites/default/files/styles/large/public/thumbnails/content-id-13592-triangular%252520slope%252520OA%252520similarity.png?itok=ZU-fXrF6)
Gradient is $\dfrac43$
AB
Image
![Triangular slope Triangular slope](/sites/default/files/styles/large/public/thumbnails/content-id-13592-triangular%252520slope%252520AB%2525201.png?itok=HecnDDSV)
Image
![Triangular slope Triangular slope](/sites/default/files/styles/large/public/thumbnails/content-id-13592-triangular%252520slope%252520AB%2525202.png?itok=tYJARJf3)
Image
![Triangular slope Triangular slope](/sites/default/files/styles/large/public/thumbnails/content-id-13592-triangular%252520slope%252520AB%252520similarity.png?itok=KDWARheR)
Gradient is $\dfrac{-3}{4}=-\dfrac34$