Friday Class 10/22 Similar Triangles

Right Triangle

Triangle ABC has a right angle at C, with |AB| = c, |BC| = a, |CA| = b. Let D be the foot of the altitude through C.

• Find |CD|, |AD|, |BD|, each as an expression in a, b, c. Explain why your answer is correct.
• If you substitute your answers above into the equation |AD| + |BD| = c, what does the equation become?

Thales figure

Let ABC be a triangle with |AB| = c, |BC| = a, |CA| = b.  Suppose that a = 6, b = 3, c = 4.

• If point E is on AC, with AE = 1, and point F is on AB with AF = 4/3, what is length EF?
• Let G be the intersection of BE and CF.  If the line parallel to BC through G intersects CA at point H, what is length AH?

Isosceles 1

Let ABC be an isosceles triangle with AB = AC. Let D be a point on AB with CD = CB.

• If |AB| = 63 and |BC| = 17, what is |BD|? Show your reasoning.

Isosceles 2

Let ABC be an isosceles triangle with AB = AC. Suppose there is a point D on segment AB so that CD = CB. Let us label the lengths: AB = AC = x and CD = CB = y.

• With these assumptions what is length BD in terms of x and y?
• Also, write the length DA in terms of x and y.

Area and similarity

If triangle DEF is similar to triangle ABC, and DE/AB = k, what ratio = (area DEF)/(area ABC)?

Construction Portfolio Part 2 (save these!)

Carry out these constructions, each one on a separate side of paper.

13.    External Tangents. Draw any circle c; label the center O and the radius r.  Then draw a point E outside the circle.  Construct two points S and T on the circle so that ES and ET are tangent to c.  Construct F as the intersection of ST and OE.

·        Let OE = d and let the radius of the circle be r. At the bottom of the page, compute the length OF as an expression in d and r.

14.    Right Triangle from Hypotenuse. Draw a segment AB and a point D on AB. Construct a point C so that ABC is a right triangle with hypotenuse AB, and D is the foot of the altitude through C.

·        Let x = |AD| and y = |BD|, then if h = |CD|, write h as an expression in x and y

15.    Geometric Mean.  Draw a segment of unit length.  Then construct a segment of length sqrt 7, using Construction 14 as the method.

16.    Half-Area. Draw a triangle ABC.  Construct points E on AB and F on AC so that EF is parallel to BC and area AEF = (1/2) area ABC