Similar/Congruent Triangle Problems
Example 1. Given that FOR is an isosceles triangle and OR || AN, find the length of AN.
Since OR || AN, we can conclude that <FRO = <FOR = <FNA = <FAN. Additionally, the vertex angle F is the same for both triangles. Since all corresponding angles in triangles FOR and FAN are congruent, these two triangles are similar.
Now we must set up ratios between corresponding sides of the similar triangles. We have:
FR/OR = FN/AN
--> 7.0/11.2 = 10.0/AN
--> AN = 10.0 * 11.2 / 7.0
--> AN = 16 units
Example 2. Find the perimeter of triangle ABC.
Since AC || RP, we know that all corresponding angles of triangles ABC and RBP are congruent, so the two triangles are similar. To find the perimeter of ABC, we must find length PB. Let x = PB. We have:
x/7 = (10+x)/(14+7)
--> 21x = 7x + 70
--> 14x = 70
--> x = PB = 5 units
We are not done! The perimeter of ABC is 18 + 10 + 5 + 7 + 14 = 54 units.
Example 3. Find the area of rectangle PLUM.
To find the area of PLUM, we must find lengths PM and PL. Since <PAL = 90 degrees, we can use the Pythagorean theorem to find length PL.
PL^2 = 6^2 + 8^2
--> PL = sqrt(36 + 64)
--> PL = 10 units
PA is an altitude of triangle PML, so triangle APL is similar to PML. We have:
AP/AL = PM/PL
--> 6/8 = PM/10
--> PM = 6 * 10 / 8
--> PM = 7.5 units
Finally the area of PLUM would be PM * PL = 7.5 * 10 = 75 units^2
Example 5. Given that triangle ABC is congruent to DEF, find x, y, and x.
We know that corresponding sides of congruent triangles are congruent, so we have:
EF = BC
--> x = 3
DE = AB
--> 2y - 3 = 4
--> y = 3.5
DF = AC
--> 3z + 2 = 5
--> z = 1
Example 5. Given that triangle ABC is congruent to triangle DCB, find x.
Since the two triangles are congruent, we know that AC || DB, so x = <ACB. To find <ACB, we have:
<ACB = 180 - 69 - 43
--> <ACB = 68 degrees
Therefore, x = <ACB = 68 degrees.