In the diagram on the left, ABCDE is
a regular pentagon. We are given
that AB= 3.61, BD = EC = 5.83 and
FD = 2.23. BE ∥ CD and 𝐴𝐸̂𝐶 =
𝐵𝐶̂𝐸 = 72° .
a) Are ∆BCD and ∆EDC
congruent?
b) Are ∆BCD and ∆EDC
similar?
c) Prove that ∆BEF and ∆CFD are
similar.
d) Find the length of BE
e) What is the length of BF and
EF, if they are equal?
f) What type of quadrilateral is
ABFE?
g) If a line was drawn, joining point A to point F, would ∆ABF and ∆AEF be similar
or congruent? Show all your working out.
h) What other lines are parallel in the diagram above? Give a reason for your
answer.
a) To determine if ΔBCD and ΔEDC are congruent, we compare their corresponding sides:
BC = ED (both are equal to 5.83)
CD = DC (common side)
BD ≠ EC (5.83 ≠ 5.83)
Since there is a pair of corresponding sides that are not equal, ΔBCD and ΔEDC are not congruent.
b) To determine if ΔBCD and ΔEDC are similar, we compare their corresponding angles:
∠BCD = ∠EDC (both are right angles)
∠CBD ≠ ∠CDE (72° ≠ 72°)
Since there is a pair of corresponding angles that are not equal, ΔBCD and ΔEDC are not similar.
c) To prove that ΔBEF and ΔCFD are similar, we need to show that their corresponding angles are equal:
∠BFE = ∠CFD (both are right angles)
∠BEF = ∠CDF (alternate interior angles)
∠BEF = ∠DTC (corresponding angles)
Since all three pairs of corresponding angles are equal, ΔBEF and ΔCFD are similar by the Angle-Angle (AA) criterion.
d) To find the length of BE, we can use the fact that BE is parallel to CD:
In ΔBAD, we have:
AB = 3.61
AD = 2 * AB (since ABCDE is a regular pentagon)
AD = 2 * 3.61 = 7.22
Using the property of ratios in similar triangles, we can set up the proportion:
(AD + DE) / BD = AB / BE
(7.22 + DE) / 5.83 = 3.61 / BE
Solving for BE:
BE = (3.61 * 5.83) / (7.22 + DE)
e) If BF and EF are equal, we can let their length be x. Then, using the property of ratios in similar triangles, we can set up the proportion:
BE / BD = BF / FD
BE / 5.83 = x / 2.23
Solving for x:
x = (2.23 * BE) / 5.83
Since BF and EF are equal, their length would be x.
f) ABFE is a trapezoid because it has one pair of parallel sides (BE and AF), and the other pair of sides (AB and EF) are not parallel.
g) If a line is drawn joining point A to point F, then we can compare ΔABF and ΔAEF:
∠ABF = ∠AEF (both are right angles)
∠BAF = ∠EAF (corresponding angles)
Since both pairs of corresponding angles are equal, ΔABF and ΔAEF are similar by the Angle-Angle (AA) criterion.
h) The other lines that are parallel in the diagram are BE and CD. This is because BE is parallel to CD and is determined by the given information (BE ∥ CD).