Study similarity of triangles using the Basic Proportionality Theorem, and apply the Pythagoras theorem in similar triangle contexts.
If a line is drawn parallel to one side of a triangle, intersecting the other two sides, it divides those two sides in the same ratio.
In triangle ABC, DE is parallel to BC, with AD=4, DB=6, AE=5. Find EC.
AD/DB=AE/EC; 4/6=5/EC; EC=7.5
A line parallel to BC cuts AB and AC at D and E. If AD/DB=2/3 and AC=15, find AE.
AE/EC=2/3, so AE=(2/5)×15=6
In triangle ABC, DE is parallel to BC, with AD=6, DB=9, AE=8. Find EC.
AD/DB=AE/EC; 6/9=8/EC; EC=12
Similar triangles have proportional sides and equal angles. The Pythagoras theorem can be proven using similar triangles formed by an altitude to the hypotenuse.
Two similar triangles have corresponding sides in ratio 3:5. If a side of the smaller triangle is 9 cm, find the corresponding side of the larger.
9×(5/3)=15 cm
Why can Pythagoras theorem be proven using similarity?
Because the altitude to the hypotenuse creates two smaller triangles similar to the original, and comparing their proportional sides algebraically yields a²+b²=c²
Two similar triangles have corresponding sides in ratio 4:9. If a side of the smaller triangle is 12 cm, find the corresponding side of the larger.
12×(9/4)=27 cm
1. In triangle PQR, DE is parallel to QR, with PD=3, DQ=6, PE=4. Find EC.
2. A line parallel to one side cuts the other two sides such that AD/DB=1/2. If AC=18, find AE.
3. Two similar triangles have side ratio 2:7. If a side of the smaller is 4 cm, find the corresponding side of the larger.
4. Are all equilateral triangles similar to each other?
5. A ladder 10m long reaches a window 8m high. Find the distance of the ladder's base from the wall.
6. In triangle PQR, DE is parallel to QR, with PD=5, DQ=10, PE=6. Find EC.
7. A line parallel to one side cuts the other two sides such that AD/DB=2/5. If AC=21, find AE.
8. Two similar triangles have side ratio 5:8. If a side of the smaller is 15 cm, find the corresponding side of the larger.
9. Are all squares similar to each other?
10. A ladder 13 m long reaches a window 12 m high. Find the distance of the ladder's base from the wall.