Q9 of 42 Page 210

AX and DY are altitudes of two similar Δ ABC and ΔDEF. Prove that AX : DY = AB : DE.


Given, Δ ABC ∼ Δ DEF


⇒ ∠ ABC = ∠ DEF


⇒ consider Δ ABX and Δ DEY


⇒ ∠ ABX = ∠ DEY


⇒ ∠ AXB = ∠ DYE = 90°


⇒ ∠ BAX = ∠ EDY


⇒ By AAA property we have in two triangles if the angles are equal, then sides opposite to the equal angles are in the same ratio (or proportional) and hence the triangles


⇒ Δ ABX ∼ Δ DEY


⇒ By Converse proportionality theorem we have


if a line divides any two sides of a triangle in the same ratio then the line must be parallel to the third side.


⇒


∴ AX:DY = AB:DE


Hence proved


More from this chapter

All 42 →
7

A flag pole 4 m tall casts a 6 m., shadow. At the same time, a nearby building casts a shadow of 24m. How tall is the building?

 8

CD and GH are respectively the bisectors of ∠ACE and ∠EGF such that D and H lie on sides AB and FE of Δ ABC and Δ FEG respectively. If Δ ABC ∼ Δ FEG then show that

i.


ii. Δ DCB ∼ ΔHGE


iii. Δ DCA ∼ ΔHGF

 10

Construct a triangle shadow similar to the given ΔABC, with its sides equal to of the corresponding sides of the triangle ABC.

 11

Construct a triangle of sides 4 cm, 5 cm and 6 cm. Then, construct a triangle similar to it, whose sides are of the corresponding sides of the first triangle.