CH-22-001 · Connected triangles: angle and shortest distance

Diagram not accurately drawn

ABC is right-angled at B. D lies on BC. AB = 15 cm, BD = 11 cm and DC = 19 cm. Calculate angle DAC and the shortest distance from B to AC. Give both answers to 2 decimal places.

SOH CAH TOA in Connected Right-Angled Triangles Booklet p. 34 & p. 37

Method

1) Use TOA (tan⁻¹) in triangles ABC and ABD and subtract for angle DAC

2) Use Pythagoras for AC and equate triangle areas ½(AB)(BC) = ½(AC)(h)

Worked Example (CH-22-001)

1) Label sides

\mathbf{H}

(Hypotenuse),

\mathbf{O}

(Opposite angle),

\mathbf{A}

(Adjacent)

2) BC = 30; angle BAC = tan⁻¹(30/15).

3) Angle BAD = tan⁻¹(11/15); subtract from angle BAC.

4) Angle DAC = 27.18°.

5) AC = √(15² + 30²).

6) Equate ½ × AB × BC to ½ × AC × perpendicular height.

7) Shortest distance = 13.42 cm.

Angle DAC = 27.18°; shortest distance 13.42 cm.