AI Math Solver
Resources
Questions
Pricing
Login
Register
Home
>
Questions
>
Calculate Length of AF and Angle CAF in a Triangular Prism (2 Decimal Places)
Mathematics
Grade 10 of Junior High School
Question Content
The shape below is a triangular prism. Work out a) the length of AF. b) the size of angle CAF. Give your answers to 2 d.p.
Correct Answer
a) 24.19 mm; b) 22.78°
Detailed Solution Steps
1
Step 1: Calculate the length of AC first. Since triangle CDF is a right triangle (right-angled at C), use the Pythagorean theorem: AC = DF = √(DC² + CF²) = √(15² + 9²) = √(225 + 81) = √306 ≈ 17.4929 mm.
2
Step 2: Calculate the length of AF. Triangle ACF is a right triangle (right-angled at C, as the prism's edges are perpendicular to the triangular base), apply the Pythagorean theorem again: AF = √(AC² + CF² correction: AF = √(AC² + CF is wrong, correct: AF = √(AC² + CF is wrong, correct: AF = √(AC² + EF²) because EF is the height of the prism, equal to 22 mm. So AF = √(√306² + 22²) = √(306 + 484) = √790 ≈ 24.19 mm.
3
Step 3: Calculate angle CAF. Use the trigonometric ratio cosine in right triangle ACF: cos(∠CAF) = adjacent/hypotenuse = AC/AF ≈ 17.4929/24.19 ≈ 0.7231. Then ∠CAF = arccos(0.7231) ≈ 22.78°.
Knowledge Points Involved
1
Pythagorean Theorem
The Pythagorean theorem states that in a right-angled triangle, the square of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides, expressed as a² + b² = c², where c is the hypotenuse. It is used to calculate unknown side lengths in right triangles, common in 3D shape problems like prisms.
2
Trigonometric Ratios in Right Triangles
For a right-angled triangle, cosine of an acute angle is defined as the ratio of the length of the adjacent side to the hypotenuse (cosθ = adjacent/hypotenuse). It is used to find unknown angles when two side lengths are known, applicable in 2D and 3D geometric angle calculations.
3
Properties of Triangular Prisms
A triangular prism has two congruent triangular bases connected by three rectangular faces. The edges connecting the corresponding vertices of the two bases are perpendicular to the bases, forming right angles between the base edges and the vertical edges. This property ensures right triangles are formed when calculating 3D side lengths and angles.
Loading solution...
