Triangular slope
Can you find the gradients of the lines that form a triangle?
Age
14 to 16
Challenge level
Problem
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What is the gradient of OA?
What is the gradient of AB?
This problem is taken from the World Mathematics Championships
Student Solutions
Answer: $\frac 43$ and $-\frac34$
Using scale factors
OA
Red triangles are similar so use scale factor $\frac35=0.6$ to find missing lengths
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Gradient is $\dfrac{4\times0.6}{3\times0.6}=\dfrac43$
AB
Green triangles are similar with scale factor $\frac45=0.8$
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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
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Gradient is $\dfrac43$
AB
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Gradient is $\dfrac{-3}{4}=-\dfrac34$