Question

1 in the figure on the right, place points a and p on line ℓ and points b and c on line m. given ℓ ∥ m, ∠abc = 90°, ab = 8 cm and bc = 10 cm, answer the questions below. (1) what kind of quadrilateral is abcp? (ans.) (2) what is the distance between lines ℓ and m? (ans.) (3) find the height when bc is the base in △pbc. (ans.) (4) find the area of △abc and △pbc. (ans.)△abc △pbc don’t forget! ●at △abc and △a’bc with side bc as the common base, if aa’ ∥ bc, then △abc and △a’bc are equal in area. 2 in the figure on the right, place points a and d on line ℓ and points b, c, e and f on line m. given ℓ ∥ m and the distance between ℓ and m is 6 cm, answer the questions below. (1) given bc = 3 cm and ef = 6 cm, find the area of △abc and △def. (ans.)△abc △def (2) find the ratio of the area of △abc to △def in this case. (ans.) : (3) given bc = 8 cm and ef = 12 cm, find the area of △abc and △def. (ans.)△abc △def (4) find the ratio of the area of △abc to △def in this case. (ans.)

Explanation:

Step1: Identify quadrilateral ABCP

Points A,P on line ℓ; B,C on line m; ℓ∥m, ∠ABC=90°. ABCP has two parallel sides (ℓ∥m) and one right angle, so it's a rectangle.

Step2: Distance between ℓ and m

AB is perpendicular to m (∠ABC=90°) and connects ℓ to m, so distance = AB = 8 cm.

Step3: Height of △PBC (base BC)

△PBC shares base BC with △ABC, and P is on ℓ∥m. Height = distance between ℓ and m = 8 cm.

Step4: Area of △ABC and △PBC

Area formula: $\frac{1}{2}×base×height$.
△ABC: $\frac{1}{2}×10×8=40$ cm².
△PBC: same height as △ABC, so $\frac{1}{2}×10×8=40$ cm².

Step5: Area of △ABC and △DEF (Problem 2(1))

Distance between ℓ and m=6 cm (height).
△ABC: $\frac{1}{2}×3×6=9$ cm².
△DEF: $\frac{1}{2}×6×6=18$ cm².

Step6: Ratio of areas (Problem2(2))

Ratio = 9:18 = 1:2.

Step7: Area of △ABC and △DEF (Problem2(3))

△ABC: $\frac{1}{2}×8×6=24$ cm².
△DEF: $\frac{1}{2}×12×6=36$ cm².

Step8: Ratio of areas (Problem2(4))

Ratio =24:36=2:3.

Answer:

1(1) Rectangle
1(2) 8 cm
1(3) 8 cm
1(4) △ABC: 40 cm²; △PBC:40 cm²
2(1) △ABC:9 cm²; △DEF:18 cm²
2(2) 1:2
2(3) △ABC:24 cm²; △DEF:36 cm²
2(4) 2:3