KCSE 2025 Maths P2 Q21 — 3D Geometry of a Cuboid, Lengths & Angles

KCSE 2025 Form 4 Geometry

Published

The Question

“A cuboid PQRSTUVW has PQ = 19 cm, QR = 16 cm and RW = 4 cm (the height). Points M and N are the midpoints of the top edges UT and VW respectively. (a) Calculate the length of line RM. (b) Calculate, correct to 2 decimal places: (i) the angle between line RM and the plane PQRS; (ii) the angle between lines RM and MQ; (iii) the obtuse angle between the planes PMNQ and MNWT.”

1

Set up coordinates

Placing the cuboid on axes turns every length and angle into a calculation. Put P at the origin with the base PQRS flat. Then Q, R and S sit on the base, and the top face is 4 cm above it. M is the midpoint of the top edge UT, so it is halfway along the 16 cm direction at the top, giving it a y of 8 and a height of 4.

P=(0, 0, 0),Q=(19, 0, 0),R=(19, 16, 0)P = (0,\,0,\,0), \quad Q = (19,\,0,\,0), \quad R = (19,\,16,\,0)
M=(0, 8, 4)M = (0,\,8,\,4)
2

Length of RM by the 3D distance formula (part a)

The distance between two points in space is the square root of the sum of the squared differences in x, y and z. Subtract the coordinates of M from R component by component, square each difference, add them and take the square root. The number under the root is a perfect square.

RM=(19−0)2+(16−8)2+(0−4)2RM = \sqrt{(19-0)^{2} + (16-8)^{2} + (0-4)^{2}}
=192+82+42=361+64+16=441= \sqrt{19^{2} + 8^{2} + 4^{2}} = \sqrt{361 + 64 + 16} = \sqrt{441}
RM=21 cmRM = 21 \text{ cm}
3

Angle between RM and the base plane (part b i)

The plane PQRS is horizontal, so the angle a line makes with it is set by how much the line rises. That rise is the vertical height, 4 cm, and the slant length is RM, 21 cm. The sine of the angle is the opposite side over the hypotenuse, so take the inverse sine of 4 over 21.

sin⁡θ=heightRM=421\sin\theta = \frac{\text{height}}{RM} = \frac{4}{21}
θ=sin⁡−1 ⁣(421)=10.98∘\theta = \sin^{-1}\!\left(\frac{4}{21}\right) = 10.98^\circ
4

Angle between lines RM and MQ (part b ii)

The two lines meet at M, so form vectors starting at M and going to R and to Q. Subtract M from each point. The angle between them comes from the dot product formula: divide the dot product by the product of the two lengths, then take the inverse cosine. Both vectors turn out to have length 21.

MR⃗=(19, 8, −4),MQ⃗=(19, −8, −4)\vec{MR} = (19,\,8,\,-4), \quad \vec{MQ} = (19,\,-8,\,-4)
MR⃗⋅MQ⃗=361−64+16=313\vec{MR}\cdot\vec{MQ} = 361 - 64 + 16 = 313
cos⁡θ=31321×21=313441\cos\theta = \frac{313}{21 \times 21} = \frac{313}{441}
θ=cos⁡−1 ⁣(313441)=44.79∘\theta = \cos^{-1}\!\left(\frac{313}{441}\right) = 44.79^\circ
5

Obtuse angle between planes PMNQ and MNWT (part b iii)

The two planes share the edge MN, which runs in the x-direction. The angle between planes is measured along lines that are each perpendicular to the shared edge. In plane PMNQ use the vector MP, and in the flat top plane MNWT use the vector MT; both are perpendicular to MN. Apply the dot product formula. The negative cosine shows the angle is obtuse, exactly what the question asks for.

MP⃗=(0, −8, −4),MT⃗=(0, 8, 0)\vec{MP} = (0,\,-8,\,-4), \quad \vec{MT} = (0,\,8,\,0)
cos⁡θ=MP⃗⋅MT⃗∣MP⃗∣ ∣MT⃗∣=−6480×8=−0.8944\cos\theta = \frac{\vec{MP}\cdot\vec{MT}}{|\vec{MP}|\,|\vec{MT}|} = \frac{-64}{\sqrt{80}\times 8} = -0.8944
θ=cos⁡−1(−0.8944)=153.43∘\theta = \cos^{-1}(-0.8944) = 153.43^\circ

Final Result

The length RM is 21 cm. The angle between RM and the base plane PQRS is 10.98 degrees, the angle between lines RM and MQ is 44.79 degrees, and the obtuse angle between planes PMNQ and MNWT is 153.43 degrees.

Why this method works

Coordinates convert 3D geometry into arithmetic: once each point has an (x, y, z), lengths follow from Pythagoras in three dimensions and angles follow from vectors. The line-to-plane angle relies on the base being horizontal, so only the vertical height lifts the line off the plane, making sine the right ratio. For two lines that meet, the dot product of vectors drawn from the meeting point directly encodes the angle between them. For two planes, the honest angle is measured perpendicular to their common edge, which is why MP and MT are chosen; a negative dot product means those directions point away from each other, so the angle between the planes is obtuse.

✓

Both MR and MQ have length sqrt(361 + 64 + 16) = sqrt(441) = 21, so the symmetric dot-product calculation cos = 313/441 gives 44.79 degrees, consistent with RM = 21 cm found in part (a).